Semiconductors and Semimetals, Volume 60 Self-Assembled InGaAs GaAs Quantum Dots

Self-Assembled InGaAslGaAs Quantum Dots SEMICONDUCTORS AND SEMIMETALS Volume 60

Semiconductors and Semimetals A Treatise

Edited by R. K. Witlardson

Eicke R. Weber

DEPARTMENT OF MATERIALS SCIENCE SPOKANE, WASHINGTON AND MINERAL ENGINEERING UNIVERSITY OF CALIFORNIA AT BERKELEY CONSULTING PHYSICIST

Self-Assembled InGaAslGaAs Quantum Dots SEMICONDUCTORS AND SEMIMETALS Volume 60 Volume Editor

MITSURU SUGAWARA OPTICAL SEMICONDUCTOR DEVICE LABORATORY FUJITSU LABORATORIES LTD. ATSUGI, JAPAN

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Contents

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LISTOF CONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1 Theoretical Bases of the Optical Properties of Semiconductor Quantum Nano-Structures Mitsuru Sugawara I . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. ELECTRONIC STATES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES . . . . . 111. INTERBAND OPTICAL TRANSITION . . . . . . . . . . . . . . . . . . . . . 1. Linear and Nonlinear Optical Susceptibility . . . . . . . . . . . . . . . 2. Spontaneous Emission of Photons . . . . . . . . . . . . . . . . . . . 3. Rate Equations for Laser Operutions . . . . . . . . . . . . . . . . . . IV . EXCITON OPTICAL PROPERTIES . . . . . . . . . . . . . . . . . . . . . . 1. State Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Effecrive-Muss Equations . . . . . . . . . . . . . . . . . . . . . . . 3. Esciton- Photon Interactions . . . . . . . . . . . . . . . . . . . . . 4. Optirul Absorption Spectra . . . . . . . . . . . . . . . . . . . . . . 5 . Spontaneous Emission of Photons in Quantum Wells and Mesoscopir Quantum Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Spontaneous Etnission of Photons in Quantum Disks Placed in a Planar Microcavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. The Coulomb Effect on Optiraf Gain Spectra . . . . . . . . . . . . . . . V . QUANTUM-DOT LASERS. . . . . . . . . . . . . . . . . . . . . . . . . 1. The Efect of Carrier Relaxation Dynamics on Laser Performance . . . . . . 2. Efect of Homogeneous Broadening of Optical Gain on Lusing Emission Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Bi-Esciton Spontaneous Emission mil Lasing . . . . . . . . . . . . . . VLSUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A P P E N D I X .., . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V

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Chapter 2 Molecular Beam Epitaxial Growth of Self-Assembled InAslGaAs Quantum Dots Yoshiaki Nakata. Yoshihiro Sugiyarna. and Mitsuru Sugawara 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. THESTRANSKI-KRASTANOW GROWTH MODE . . . . . . . . . . . . . . . 1. Energy-Balmce Model.for Islund Formation . . . . . . . . . . . . . . 2. InAs Island Growth . . . . . . . . . . . . . . . . . . . . . . . . . 3. Multiple-Layer Growth and Perpendicular Alignment ofls1and.r. . . . . . 4. In-Plane Alignment of Isbnds . . . . . . . . . . . . . . . . . . . . . 111. CLOSELY STACKED InAs/GaAs QUANTUM DOTS. . . . . . . . . . . . . . 1. Close Stacking ofInAs Islands . . . . . . . . . . . . . . . . . . . . . 2. Photoluminescence Properties . . . . . . . . . . . . . . . . . . . . . 3 . Zero-Dimensional Exciton Confnmwnt Ewluuted by Diumagnetic ShiJis . . IV. COLUMNAR InAsJGaAs QUANTUM DOTS. . . . . . . . . . . . . . . . . V . SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . .

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117 119 119 121 125 130 132 133 137 140 143 150 151 152

Chapter 3 Metalorganic Vapor Phase Epitaxial Growth of Self-Assembled InGaAslGaAs Quantum Dots Emitting at 1.3 pm Kohki Mukai. Mitsuru Sugawara. Mitsuru Egawa. and Nobuyuki Ohtsuka I. I1. 111. IV .

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . ATOMICLAYER EPITAXIAL GROWH. . . . . . . . . . . . . . . . . . . . ALTERNATESUPPLY GROWTH OF InGaAs DOTSBY In-As-Ga-As SEQUENCE. . . ALTERNATESUPPLY GROWTH OF InGaAs DOTSBY THE In-Ga-As SEQUENCE . . . 1. Two Types o f A L S Dot . . . . . . . . . . . . . . . . . . . . . . . . 2. Multil,l e-Layer Growth . . . . . . . . . . . . . . . . . . . . . . . . V . THEGROWTH PROCESS. . . . . . . . . . . . . . . . . . . . . . . . . VLSUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 157 160 166 168 172 176 180 181

Chapter 4 Optical Characterization of Quantum Dots Kohki Mukai and Mitsuru Sugawara I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. LIGHTEMlSSlON FROM DISCRETE ENERGY STATES . . . . . . . . . . . . . . 1. Photoluminescence Photoluminescence Excitation. und Electroluminescence Spectru . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Wufer Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Microprobe Photoluminescence . . . . . . . . . . . . . . . . . . . . 111. CONTROLLABILITY OF QUANTUM CONFINEMENT . . . . . . . . . . . . . . . 1. Two Methods of Controlling Quuntizud Energies . . . . . . . . . . . . . 2. Magneto-Optical Spectroscop!. . . . . . . . . . . . . . . . . . . . . IV. RADIATIVEEMISSION EFFICIENCY . . . . . . . . . . . . . . . . . . . . . V . SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 5 The Photon Bottleneck Effect in Quantum Dots Kohki Mukai and Mitsuru Sugawara I . INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . A MODELOF THE CARRIER RELAXATION PROCESS IN QUANTUM DOTS. . . . . . I11. EXPERIMENTS ON LIGHTEMISSION AND CARRIER RELAXATION I N QUANTUM-DOT DISCRETE ENERGYLEVELS. . . . . . . . . . . . . . . . . . . . . . . . 1. Electrolurninescence Spectra . . . . . . . . . . . . . . . . . . . . . . 2. Time-Resolved Photoluminescence . . . . . . . . . . . . . . . . . . . 3 . Simulation of Electroluminrscencr Spectra . . . . . . . . . . . . . . . . IV . INFLUENCE OF THERMAL TREATMENT. . . . . . . . . . . . . . . . . . . I . Change in Emission Spectra ufier Annealing . . . . . . . . . . . . . . . 2. Competition between Carrier Relasation and Recombination . . . . . . . . V. SIMULATION OF LASER~ R F O R M A N C EINCLUDING THE AUGER RELAXATION PROCESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VLSUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 6 Self-Assembled Quantum Dot Lasers Hajime Shoji I . INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. FUNDAMENTAL PROPERTIES OF QUANTUM-DOT LASERS. . . . . . . . . . . . 1. Gain Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 2. Threshold Current . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Dynamic Characteristics . . . . . . . . . . . . . . . . . . . . . . . 111. FABRICATIONOFSELF-ASSEMBLEDQUANTUM-DOTLASERS . . . . . . . . . . 1. Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Device Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 3. Limiting Factors of laser Performance . . . . . . . . . . . . . . . . . IV . KEYTECHNOLOGIES FOR THE NEXTERA . . . . . . . . . . . . . . . . . . 1. Closely Stacked Quantum-Dot Lasers . . . . . . . . . . . . . . . . . . 2. Columnar Quantum-Dot Lasers . . . . . . . . . . . . . . . . . . . . 3. Lorig- Wavelength Quantum-Dot Losers . . . . . . . . . . . . . . . . . 4 . Quantum-Dot Vertical-Cuvit.v Sur:fiice-EmittingLasers . . . . . . . . . . V. CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241 243 244 246 248 249 250 255 267 269 270 273 276 279 282 283 283

Chapter 7 Applications of Quantum Dot to Optical Devices Hiroshi Ishikawa I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . I1. PROPERTIES OF QUANTUM DOTS . . . . . . . . . . . . . . . . . 1. The Quantuni Dot us a Two-Level System . . . . . . . . . . 2. Attractive Features of Quantum Dots.for Device Application . . . I11. QUANTUM DOTSFOR VERYHIGH SPEEDLIGHTMODULATION . . . . 1. The Needfor High-Speed. Low- Wuwlength-Chirp Light Sources .

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2. Direct Modulation of Quantum-Dot Lasers . . . . . . . . . . . . . . 3. The Quantum-Dot Intensity Modulator . . . . . . . . . . . . . . . . IV . QUANTUM DOTSAS A NONLINEAR MEDIUM . . . . . . . . . . . . . . . 1. The Need for Large Nonlinearity with a Large Bandwidth . . . . . . . . 2. Analysis of xf3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V . PERSISTENT HOLEBURNING MEMORY. . . . . . . . . . . . . . . . . . 1. Persistent Spectral Hole Burning Memory Using Quantum Dots . . . . . 2. Experimenral Results . . . . . . . . . . . . . . . . . . . . . . . . 3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . SUMMARY AND PERSPECTIVES ON QUANTUM-DOT OPTICAL DEVICES. . . . . 1. Trends in Optoelectronics . . . . . . . . . . . . . . . . . . . . . . . 2. Uses for Quantum-Dot Optical Devices . . . . . . . . . . . . . . . .

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ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

298 302 303 303 306 311 314 314 316 319 319 320 321 321 321

Chapter 8 The Latest News Mitsuru Sugawara. Kohki Mukai. Hiroshi Ishikawa. Koji Otsubo. and Yoshiaki Nakata I. LASING WITH LOW-THRESHOLD CURRENT AND HIGH-OUTPUT POWER FROM COLUMNAR-SHAPED QUANTUM DOTS . . . . . . . . . . . . . . . . I1. EFFECTOF HOMOGENEOUS BROADENING OF SINGLE-DOT OPTICAL GAIN ON LASING SPECTRA. . . . . . . . . . . . . . . . . . . . . . . . . . 111. QUANTUM DOTSON INGAASSUBSTRATES . . . . . . . . . . . . . . . . IV . QUANTUM DOTSEMITTING AT 1.3 pm GROWN BY Low GROWTH RATES AND WITH AN INGAASCAP . . . . . . . . . . . . . . . . . . . . . . . V . REDUCED-TEMPERATURE-INDUCED VARIATIONOF SPONTANENOUS EMISSION IN ALTERNATESUPPLY (ALS) QUANTUM DOTSCOVERED BY In,.,Ga, ,As . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Preface This volume is devoted to the crystal growth, optical properties, and optical-device applications of self-assembled InGaAs quantum dots -one of the major areas of semiconductor research today. The quantum dots’ atom-like density of states should significantly improve optical performance, especially in semiconductor lasers, and contribute to the development of new optical devices. However, because of the difficulty of their fabrication, quantum-dot optical devices, first posited by Arakawa in 1982, were little more than a dream until the advent of self-assembled quantum dots grown via molecular beam epitaxy (MBE) and metal organic vapor phase epitaxy (MOVPE) in the early nineties. Since then, the field has grown rapidly. The performance of quantum-dot lasers has now reached that of the wellestablished quantum-well lasers, and further progress is expected in the near future. The authors, all of whom work for Fujitsu Laboratories in Atsugi, Japan, are leaders in quantum-dot research for optical device applications. They have not only achieved room-temperature quantum-dot laser operation with low threshold current and high efficiency but have also gained wide knowledge of optical device designs and processes, laser properties, device simulation, theoretical aspects of optical characteristics, MOVPE and MBE growth techniques, and optical and crystallographic characterization. Here, the Fujitsu group provides both general knowledge and state-of-the art research results. They also discuss additional research needed to realize high-performance optical devices. For these reasons, this volume not only represents a milestone in semiconductor technology, treating many topics not been treated elsewhere; it also contributes considerably to the field’s further development. Chapter 1 provides a basic theoretical background for the optical properties of semiconductor quantum nano-structures in particular, those of quantum wells and quantum dots and the differences between them. Also covered are interband optical transition, exciton optical properties, and quantum-dot laser simulations primarily focusing on carrier relaxation. ix

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Experimental results and numerical calculations are included to improve understanding of the derived theoretical expressions. The formulae are presented step by step, without neglecting numerical coefficients, and employ the MKSA system of units. This should help readers calculate the optical response of semiconductor nano-structures on their own. Chapter 2 focuses on the molecular beam epitaxial growth of selfassembled InAs islands on GaAs substrates through the Stranski-Krastanow (SK) growth mode. These islands work as quantum dots because they are smaller than the exciton optical Bohr radius in this material system. The authors’ method of stacking islands perpendicularly creates closely stacked and columnar-shaped quantum dots. The perpendicular stacking increases the dot size because of the electronical coupling of each stacked island in the growth direction, enabling tuning of the emission wavelength as well as narrowing of the spectrum line width caused by the island-size fluctuations. A comparison of the structural optical properties of different types of dots proves the advantages of the columnar-shaped type. These advantages are demonstrated in Chapters 6 and 8. Chapter 3 introduces another original type of self-assembled quantum dot, grown by MOVPE, whose growth sequence is unique in that the group-111 and group-V precursors are supplied alternately with an amount corresponding to one monolayer or less. This explains the dot’s name: ALS, which stands for “alternate supply.” One of the most striking features of ALS dots is that they emit at a wavelength of 1.3 pm at room temperature, which is the zero-dispersive wavelength of silica optical fiber used in the optical data transmission system. In addition, their emission spectrum linewidth is very narrow and provides a series of distinct lines peculiar to three-dimensional quantum confinement. These two features make ALS dots very attractive for practical application. Chapter 4 deals with optical characterization of quantum dots, focusing primarily on the ALS type. Through various diagnostic techniques, the unique properties of ALS dots are presented, including long emission wavelength, emission spectra with multiple peaks from discrete energy states, harmonic-oscillator type confinement potential, large wavelength tunability between 1.2 and 1.5 pm through size control, and carrier lifetimes through radiative and nonradiative recombinations. Chapter 5 presents experimental studies on the carrier dynamics in selfassembled InGaAs/GaAs quantum dots. The physics of carrier relaxation in quantum dots has been studied intensively ever since quantum dots were discovered to have promising features for optoelectronic device application. One of the primary concerns of many researchers has been whether carrier relaxation into quantum-dot discrete states is significantly slowed because of the lack of phonons needed to satisfy the energy conservation rule. This is known as the phonon bottleneck and at one time was considered merely

PREFACE

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a theoretical problem; however, the recent advent of self-formed quantum dots has enabled researchers to evaluate it experimentally. The chapter provides evidence of retarded carrier relaxation and discusses how the phonon bottleneck works in quantum dots. In addition, the discussion on how retarded carrier relaxation affects quantum-dot laser performance given in Chapter 1 is reopened here, taking into account the Auger relaxation process. Chapter 6 describes semiconductor lasers with self-assembled quantum dots in the active layer. After a brief theoretical discussion of what can be expected from quantum-dot lasers, the chapter moves on to their presentstage performance as well as their fabrication processes, lasing characteristics, and problems to be solved. Finally discussed are several key technologies for future improvement of quantum-dot lasers, including new trends such as closely-stacked and columnar-shaped structures, long wavelength emission, and vertical-cavity surface-emitting lasers (VCSELs). Chapter 7 reviews some of the useful properties of quantum dots and how they might apply to laser technology. Problems associated with the conventional technologies for long-distance high-bit-rate communication systems are reviewed, as are potential solutions such as direct modulation of quantum dot lasers and the feasibility of external modulators using quantum dots. To determine the feasibility of quantum dots as a nonlinear medium, the authors perform a trial analysis of the dots’ third order nonlinear susceptibility. They also discuss the use of quantum dots for high-density optical memory through persistent spectral hole burning, and, finally, the role of quantum-dot-based devices in future optoelectronics. Chapter 8 presents the latest developments in quantum-dot research: lasing with low-threshold current and high-output power from columnarshaped quantum dots; the effect of homogeneous broadening of single-dot optical gain on lasing spectra; quantum dots on InGaAs substrates; quantum dots emitting at 1.3 pm grown by low growth rates and with an InGaAs cap; and reduced-temperature-induced variation of spontaneous emission wavelength in ALS quantum dots covered by In,,,Ga,,,As. The success of this volume, as well as the success of the field, will be largely due to its outstanding contributors. The authors wish to thank Dr. Hajime Ishikawa, Dr. Naoki Yokoyama, Dr. Shigenobu Yamakoshi, Dr. Hajime Imai, and Dr. Kiyohide Wakao of Fujitsu Laboratories Ltd. for their encouragement and strong support. As volume editor, I wish to thank Miki Sugawara for her invaluable practical support for this volume and her continuous encouragement during my research. MITSURU SUGAWARA

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List of Contributors

Numbers in parenthesis indicate the pages on which the authors’ contribution begins.

MITSURUEGAWA,(1 55) Optical Semiconductor Devices Laboratory, Fujitsu Laboratories Limited, Atsugi, Kanagawa, Japan HIROSHI ISHIKAWA, (287, 325) Electron Devices and Materials Laboratory, Fujitsu Laboratories Limited, Atsugi, Kanagawa, Japan KOHKIMUKAI,(155, 183, 209, 325) Optical Semiconductor Devices Laboratory, Fujitsu Laboratories Limited, Atsugi, Kanugawa, Japan YOSHIAKI NAKATA, (1 17, 325) Quantum Electron Devices Laboratory, Fujitsu Laboratories Limited, Atsugi, Kanagawa, Japan NOBUYUKI OHTSUKA, (1 55) Integrated Materials Laboratory, Fujitsu Laborutories Limited, Atsugi, Kanugawa, Japan

KOJI OTSUBO,(325) Electron Devices and Materials Laboratory, Fujitsu Laboratories Limited, Atsugi, Kanagawa, Japan HAJIMESHOJI,(241) Optical Semiconductor Devices Laboratory, Fujitsu Laboratories Limited, Atsugi, Kanagawa, Japan MITSURUSUGAWARA, (1, 117, 155, 183, 209, 325) Optical Semiconductor Devices Laboratory, Fujitsu Laboratories Limited, Atsugi, Kanagawa, Japan YOSHIHIRO SUGIYAMA, (1 17) Quantum Electron Devices Laboratory, Fujitsu Laboratories Limited, Atsugi, Kanagawa, Japan

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SEMICONDUCTORS AND SEMIMETALS. VOL. 60

CHAPTER 1

Theoretical Bases of the Optical Properties of Semiconductor Quanturn Nano-St ructures Mitsuru Sugawara OPTICAL SEMICONDUCTOR DEVICES LABURATORY

FUJITSULABORATORES LTD. ATSUGI.KANAGAWA, JAPAN

I. INTRODUCTION . . . . . . . . . . .

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11. ELECTRONIC STATES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES . . . .

111. INTERBAND OPTICAL TRANSITION . . . . . . . , . . . , . , . . , , 1. Linear and Nonlinear Opticul Susceptibiliiy . . . . . . . . . . . . . 2. Spontaneous Emission of Photons . . . , . . . , , . , . . , . , . 3. Rate Equations for Laser Opcrutions . , . . . . , , . . . , . . . IV. EXCITON OPTICAL PROPERTIES . . . . . . , . . . , , . . . . . . . 1. State Vectors . . . . . . . . . . . . , . . . . . . . . . . . . 2. Effective-Mass Equations . . . . . . . . , , . . . . . . . . . . 3. Exciton-Photon Interactions . . . . . . . , , . . . . . . . . . . 4. Opticul Absorption Spectra . . . . . , . , . . . . . . . . . . . 5. Spontaneous Emission of Photons in Quantum Wells and Mesoscopic Quantum Disks . . . . . . . . . . . . . . . . . , . . , , . . 6. Spontaneous Emission of Photons in Quantum Disks Placed in a Planar Microcavity . . . . . . . . . . . . . . . . . . . . . . . . . . I. The Coulomb Effect on Opticd Guin Specisu . . . . . . , , . . , . V. QUANTUM-DOT LASERS . . . . . . . . . . . . . . . . . , . , , , 1. The Effect of Carrier RdcJXUliOn Dynunnc.s on Luser Per:/Om~nnce . . . 2. Efect of Homogeneous Brmlrning of Opticul Gain on Lasing Emission Spcctsu.. . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Bi-Exciton Spontaneous Emission unrl Lasing . . . . . . . . . . . . VI. SUMMARY. . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix.. . . . . . . . . . . . . . . . . . . . . . . . . . , , . . References . . . . . . . . . . . . . . . . . . . . . . . . , . . . . .

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I. Introduction Research on synthesized semiconductor quantum nano-structures was initiated by the discovery of superlattices by Esaki and Tsu (1969, 1970). These researchers observed quantum effects in one-dimensional periodic 1 Copyright I 1999 by Academic Press All rights of iepioduLtron in any lorm reserved ISBN 0-12-752169-0 ISSN 0080-8784!99 $70 00

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MITSURUSUGAWARA

structures consisting of alternating, differing layers, whose thickness was less than that of the electron mean free path. They named their periodic structure a semiconductor superlattice and predicted negative resistance and Bloch oscillations for electrons moving toward the superlattice potential. Subsequent experimental proof of the negative resistance (Chang et al., 1974) and the finding of enhanced exciton resonance in quantum wells (Dingle, 1975) had a great impact on the development of this research field. The concept rapidly became the standard for producing and designing semiconductor multilayers with desirable electronic and optical properties. Semiconductor quantum wells, in which a thin semiconductor film is sandwiched between different materials via heterojunctions, confine electron motion in the two-dimensional thin-film plane. This two-dimensionality gives rise to new optical properties not observed in bulk materials, such as optical absorption and gain spectra peculiar to the steplike density of states (Weisbuch, 1987), strong exciton resonance clearly observable even at room temperature (Ishibashi et al., 1981; Miller et al., 1982a), large optical nonlinearity (Miller et al. 1982b, 1983; Chemla et al., 1984) and an electric field-induced energy shift of the resonance, called the quantum-confined Stark efect (Chemla et al., 1983; Miller et al., 1984a, 1985a). These properties led the way to a variety of new optical devices: quantum-well lasers (Tsang, 1981), high-speed optical modulators (Wood et al., 1984), optical switches (Miller et al., 1984b), optical bistable devices such as self-generated electrooptic effect devices (SEED’S) (Miller et al., 1985b) and the like. Quantumwell lasers and modulators, in particular have become conventional, widely used devices in optical transmission systems and optical data storage. Multi-dimensional quantum-confinement structures, such as quantum wires and quantum dots, are expected to further improve quantum-effect optical devices. Their most valuable application will be quantum-dot and quantum-wire lasers, which will be far superior to quantum-well lasers, as predicted by Arakawa and Sakaki (1982). This is primarily because, as the dimension of materials is reduced, the density of states for electrons changes to give a narrower optical transition spectrum, making the light-matter interaction more efficient. In line with the prediction of Arakawa and Sakaki, great efforts have been made to form lateral confinement potentials in addition to quantum wells, mostly using high-resolution photo- or electron-lithography and crystal growth on masked or etched substrates (see references in Chapter 2). However, crystals fabricated by these artificial processes have suffered from structural problems such as being too large to produce observable quantum effects and having low crystal quality at the interfaces. A breakthrough occurred in the early 1990s with the formation of self-assembled quantum dots during the highly strained growth processes of molecular beam epitaxy

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES3

(MBE) and metal-organic vapor phase epitaxy (MOCVD), which created high-quality quantum dots suitable for optical devices (see Chapters 2 and 3). The atom-like density of states in quantum dots has been observed (see Chapter 4) and room-temperature quantum-dot laser operations have been realized (see Chapter 6). Promising applications of quantum dots to other optical devices are discussed in Chapter 7. We are now entering a new world of zero-dimensional physics and devices, after almost 25 years of research on two-dimensional quantum wells. This chapter provides a basic theoretical background for the optical properties of semiconductor quantum nano-structures. One of its primary purposes is to show how the optical properties of quantum dots differ from those of quantum wells. The chapter is written for students who have finished a course on solid-state physics as well as professional researchers. In particular, experimental researchers who are struggling with difficult papers and textbooks written by theorists should benefit from the information provided here. The formulae are presented step by step and without neglecting numerical coefficients; moreover, the MKSA system of units is used to help readers calculate the optical response of semiconductor nano-structures by themselves. Experimental results and numerical calculations are included to improve understanding of the derived theoretical expressions.

11. Electronic States of Semiconductor Quantum Nano-Structures

Recent progress in nano-scale growth techniques such as MBE and MOCVD has enabled us to grow high-quality semiconductor quantum wells and quantum dots. Figure 1.1 shows cross-sectional transmission electron microscope (TEM) photographs of (a) In,,,,Ga,~,,As/InP strained quantum wells and (b) self-assembled In,.,Ga,.,As/GaAs quantum dots, both grown by MOCVD. The quantum well consists of a 9.7-nm In,,,,Ga,~,,As well layer sandwiched by InP. It is lattice-mismatched to InP with an in-plane strain of -0.81%. Electrons in the conduction band and holes in the valence band are spatially confined in the In,.,,Ga,,,,As, which acts as a potential well; that is, the quantum well has a type-I band lineup. The self-assembled quantum dots are grown by our original process to alternately supply InAs and GaAs monolayers. The InAs supplied to the GaAs substrate has about 7% lattice mismatch to the GaAs substrate, leading to the self-assembling of In,~,Ga,,,As clusters instead of a film to relax strain energy (see Chapters 3 and 4). These dots differ from conventional selfassembled In(Ga)As/GaAs quantum dots grown via the Stranski-Krastanov

4

MITSURUSUGAWARA

InP

InGaAs

20 nm H

FIG. 1.1. Cross-sectional transmission electron microscope (TEM) photograph of (a) In,,,,G,,,,As/InP strained quantum wells. and (b) self-assembled In, ,Ga,,,As/GaAs quantum dots, both grown by MOCVD. The self-assembled quantum dots are grown by our original growth sequence of alternately supplying InAs and GaAs monolayers (see Chapter 3).

(SK) mode. In particular, they emit at a much longer wavelength of 1.3 pm at room temperature- a suitable wavelength for optical communication systems- and they are often accompanied by a thicker quantum-well surrounding layer, whose counterpart in the SK growth mode is a wetting layer. Quantum nano-structures and their density of states are illustrated in Fig. 1.2. When the size of the crystal is reduced to the nanometer scale in one direction and the crystal is surrounded by other crystals acting as potential barriers, the freedom of electron movement is lost in that direction. The potential height corresponds to the band offset between the two crystals in the conduction and valence bands. Electrons in the quantum well move in the x-y plane; those in the quantum wire move in the x direction; and those in the quantum dot are completely localized. This confinement results in the quantization of the electron energy and in the variations of the electron density of states, which is the most remarkable and significant benefit of low-dimensional semiconductor technology. In this section, we briefly survey how the energy, the wave function, and the density of states depend on the dimension of quantum nanostructures, using an elementary quantum-mechanical approach. The effective-mass approximation effectively describes the electronic state of bulk semiconductors. In semiconductor quantum wells, this approximation has also succeeded in calculating quantum energy shifts as a function of the well-layer width, as well as various optical properties arising from the quantized states. The main assumption of the effective-mass approximation is that the envelope wave function does not vary a great deal in the unit cell with a length scale of subnanometers; this assumption holds up in quantum nano-structures like those in Fig. 1 .l. Assuming a parabolic band dispersion,

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES5 Bulk

r3L

Quantum wire

V=L3

N,: Areal density

=t

Energy

Energy

X Quantum well

Quantum dot

OW

L Well

Barrier

D = L2

Energy

Energy

FIG. 1.2. Schematic view and graph of quantum nano-structures and their density of states.

band-edge electron states of semiconductors can be described by the effective-mass equation as (Haken, 1973, for example)

1

V2 + V(r) Fk(r) = EFk(r) Here, m* is the effective mass; h is the Planck's constant divided by 2.n; r = (x,y, z ) is the electron position vector; V(r) is the confinement potential, Fk(r) is the envelope wave function; and E is the electron energy. Using the periodic part of the band-edge Bloch function normalized in the unit cell, uo(r), the normalized electron wave function close to the band edge is written simply as

6

MITSURUSUGAWARA

where s2 is the unit-cell volume. The 111-V and 11-VI semiconductor materials have s-like conduction-band-edge states and p-like valence-bandedge states that consist of the heavy-hole state, the light-hole state and the spin split-off state under spin-orbit interactions. See the appendix to this chapter for uo in the conduction and valence bands. According to the k * p perturbation theory, the band mixing of these states determines the effective mass, that is, the band dispersion of the conduction and valence bands (Kane, 1956, 1957). In the k - p calculation, the electron wave function is expanded on the basis of the band-edge state functions, and the eigenvalue problem is solved by taking into account the k - p terms up to the second order. In the design of state-of-art quantum-well devices, such as strained quantum-well lasers, a knowledge of band mixing and resultant band nonparabolicity is indispensable in engineering optimal band structures for high performance (Bastard et al., 1991, Chuang, 1991, Sugawara et al., 1993a, 1994). Readers interested in this topic should see Chuang’s well-written textbook (1995). This so-called band engineering will also be done in quantum dots if we can learn how to control the dot’s fabrication. Equation (1.1) uses a constant and isotropic effective mass, whether the band is conduction or valence, to uncover the primary role of low-dimensional quantum confinement. Assuming barrier potentials with infinite height, the eigenenergies, the envelope wave functions, and the density of states are given as follows. Bulk Materials

Setting V(r) = 0, Eq. (1.1) gives the energy of E =

E(k) =

h2k2 2m*

~

and the envelope function of

-

1 Fdr) = -exp( - i k r)

Jv

(1.4)

where V = L3 is the crystal volume. The envelope wave function is normalized in the crystal. The wave vector of k = ( k x ,k,, k,) satisfies the periodic boundary conditions as k , = (2nn,)/L, k, = (2nn,)/L, and k, = (2nn,)/L; n x , n y , and n, are integers. The density of states per unit volume, which is the number of states between the energy of E and E + dE, is given as

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES7

D(E) = hxG[E(k) Vk -- x-

v

'

(243

-

El

x R,

[:

dkk2G[E(k) - El

Here, R, = 4n is the integration in the angular part. The factor of two represents the degeneracy due to spin.

Quantum Wells The confinement potential for the square quantum well is given as V(r) = V(z), where V ( z ) = co when IzI 3 L,/2; V(z) = 0 when IzI < L,/2; and L, is the well width (Fig. 1.3(a)). Under the infinite potential height, the wave function vanishes at the boundaries between the well and barrier layers. The eigenvalue is

with the corresponding normalized wave function of 1 Fk(r)= -cp,,(z)

@

exp(--ikll*rll)

and

Here, D = L2 is the area of the quantum well; k = (k", n,n/L,); the in-plane wave vector is kll = ( k x ,k y ) ; r = (rll, z); rll = (x,y); and n, = 1,2,3,. . . . The wave function in the z direction becomes stationary. The minimum energy and the energy separation between each quantized state increase as the well width decreases. Figure 1.3(b) illustrates the parabolic dispersion curve in the conduction and valence bands. The conduction-band electron effective mass is written as mQ. The superscript 11, indicates that the mass is for the in-plane carrier movement. Since the valence-band effective mass is negative in most cases, it is more convenient to introduce a positive particle of a hole

8

M~TSURU SUGAWARA

'T

Et Conduction

transition

________----____---

(4

I I

band

(b)

I

FIG. 1.3. (a) Confinement potential for the square quantum well, V ( z ) , and the quantized energies for an electron in the conduction band. En*-,and for a hole in the valence band, En&=. E, is the band gap of the well material. (b) The band dispersion of the conduction band and valence band in the quantum-well plane. mr" is the electron effective mass, mrll is the hole effective mass, and m:I1 is the reduced mass. Interband optical transition occurs almost perpendicularly in the dispersion curve.

with the opposite effective mass of m;. Then, Eq. (1.6) holds also for holes with the energy origin at the valence-band maximum and with the energy positive in the downward direction. EneZ(or E,J and En,= (or En,J in Fig. 1.3(a) are quantized energies for electrons and holes, respectively. The quantized wave functions are described as cpne,(z) (or cp,,Jz)) for electrons, and cp,,,(z) (or cp.,,(z)) for holes. Letting LQW be the quantum-well thickness, that is, the sum of the well and barrier regime thickness- the density of states per unit volume is

where R, = 2x is the integration in the angular part and O(x) is the Heaviside step function (O(x) = 1 for x 2 0 and O(x) = 0 for x < 0). As shown in Fig. 1.2, the density of states is steplike.

1

OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES9

Quan rum Wive The confinement potential for the square quantum wire is given as V(r) = V(y) V(z), where V(y) is the additional confinement potential, with V ( y ) = GO under lyl 3 L,/2 and V ( y ) = 0 under lyl < LJ2; and L, is the y-direction length of the wire cross-section. The eigenvalue to the infinite barrier model is

+

c = E ( k ) = ,h2k2 ,.En,,+En==~[k:+(~)2+~~)2] 2m*

(1.10)

where EnYhas the same form as that of En=of Eq. (1.6) and ny,II,= 1,2,3.. . . The electron energy further increases due to the additional confinement. The corresponding normalized wave function is

Here, L is the length of the quantum wire, and k = ( k x ,nyn/Ly,n,n/L,). Letting NWi be the area density of the quantum wires (the number of quantum wires divided by the quantum-wire region area in the y-z plane), the density of states per unit volume is 2N

D(E)

.

G[E(k) - E l ny,n,.k,

2Nwi

L

L

2n

Nwi n

J2m* c

=--X-XQ1

-

where Q,

~~

h

1

1;

dkG[E(k)

- El

ny,nz

ny.nz

1

J E- E . ~ E,,=

[eV-

K~]

(1.12)

= 2.

Quantum Dots Assuming an infinitely high potential for all directions, the confinement potential becomes V(r) = V ( x ) V ( y ) + V(z). Thus, we get

+

E =

E(k)

=

+ EnY+ En== __ 2m*

Enx

h2

[(y)’+ (?>’+ (y)’]

(1.13)

10

MITSURUSUGAWARA

where n,, ny,n, = 1,2,3.. . and k = (n,n/L,, nyz/Ly,n,z/L,). The energy states are completely discrete. The corresponding wave function is

Letting N , be the volume density of quantum dots, the density of states is a series of delta functions as

The interband optical transition between the conduction band and the valence band reflects the density of states derived above. Figure 1.4(a) shows the optical absorption spectra of the quantum well of Fig. l.l(a) (Sugawara, 1993a). The absorbance is measured by (1.16) where li, is the intensity of the incident light beam; I,, is that of the transmitted light beam; and N , is the number of well layers-10 in this case. Clearly observable is the steplike optical absorption spectrum due to the two-dimensionality of quantum wells. (Strictly speaking, this clear steplike absorption continuum arises from the parabolic nature of the ground-state heavy-hole valence band, which is caused by enhanced splitting against the light-hole valence subband under the in-plane compressive strain.) In addition, we see peaky spectra due to the electron-heavy-hole exciton resonance (le-hh) at the absorption edge and the electron-lighthole resonance (le-lh) on the absorption continuum at a shorter wavelength. We also observe higher-order electron-heavy-hole exciton resonance, as (2e-hh). The numbers of 1 and 2 represent the z-direction quantum number, nz, which should be equal between the conduction and valence bands, for the allowed optical transition (see the appendix). Figure 1.4(b) shows the photoluminescence spectra of the quantum dots of Fig. l.l(b) (Mukai and Sugawara, 1999). As the excitation laser power increases, peaky emissions appear one by one, caused by the interband transition between the discrete energy levels of the conduction and valence bands. This is experimental proof of the zero-dimensionality of electrons and holes in quantum dots. In the following sections, formulae to describe the optical response of quantum nano-structures are derived.

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 11

Pe-hh 0 t

-

1e-hh le-lh

0 1.2

(4

1

(b)

1.6

1.4

Wavelength (pm)

1.2

1.4

1.6

Energy (eV)

FIG. 1.4. (a) Optical absorption spectrum at 77K of the In,,,Ga,,,,As/InP strained quantum wells of Fig. l,l(a). The steplike optical absorption spectrum due to the twodimensional character of the density of states is clearly observed. Resonant spectra are due to the ground-state electron-heavy-hole exciton (le-hh) at the absorption edge, the electronlight-hole exciton (le-lh) on the absorption continuum at the shorter wavelength, and the higher-order electron-heavy-hole exciton (2e-hh) (from Sugawara et al., 1993a. Copyright 1993 by The American Physical Society). (b) Photoluminescence spectra at 5 K of the In,,,Ga,,,As/GaAs quantum dots of Fig. l.l(b) for various excitation level. Emissions from the discrete energy levels due to three-dimensional quantum confinement are clearly observed (from Chapter 4).

111. Interband Optical Transition The interband transition of an electron between the conduction and valence bands occurs by the transfer of the interband energy between the electron and the photons. When semiconductors have a direct band gap, the optical transition between the band edges occurs at a highly efficient rate,

12

MITSURUSUGAWARA

making the materials quite useful for optical devices. “Direct” means that both the conduction-band minimum and the valence-band maximum are at the same electron wave vector position in the band dispersion diagram, making possible the electron-hole recombination by the emission of photons, with a wave vector selection rule satisfied. This is also the case in quantum nano-structures consisting of direct band-gap materials and having type-I band lineup, where both electrons and holes are confined in the materials. A variety of material systems fall into this category, such as GaAs/AlGaAs, InGaAsP/InP, InGaP/AlGaInP, and InGaN/GaN. The optical responses of semiconductors are categorized as optical absorption, stimulated light emission, spontaneous light emission, and various nonlinear optical processes. All are applied to various optical devices-for example, optical absorption to photodetectors, modulators, and solar batteries; stimulated emission to lasers and optical amplifiers; spontaneous emission to light-emitting diodes; nonlinear processes to switches and wavelength conversion devices. Depending on the required photon energy in practical application fields such as optical transmission systems and optical storage, we choose appropriate semiconductor materials, from infrared to red to blue to violet. As seen in Fig. 1.4, the optical response varies, depending on the confinement dimension. This section presents the theoretical bases of the interband optical transitions in quantum nano-structures. In addition, a prescription to simulate lasing operation using rate equations is briefly summarized. The excition effect due to the electron-hole Coulomb interaction is not taken into account here, but is treated in Section IV. By comparing the results in this section with those in Section IV, readers will see how the exciton effect influences semiconductor optical response.

1. LINEARAND NONLINEAR OPTICALSUSCEPTIBILITY Let us derive linear and nonlinear optical susceptibilities using the density-matrix theory, which plays an important role in any discussion of the optical properties of materials in quantum electronics. The optical absorption and gain coefficient can be derived from the susceptibilities. A great merit of using the density-matrix approach is that the dephasing of light-induced polarization can be incorporated, giving us the spectrum profile of the linear and nonlinear response. Chapter 7 presents the calculation of the dephasing rate due to electron-electron and hole-hole scattering in quantum dots. Readers unfamiliar with the density-matrix approach should refer to the quantum optics textbook of Meystre and Sargent 111 (1990) and other similar works.

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 13

By taking )I as the electron wave function of the material system and expanding it using the complete sets of the conduction-band and the valence-band states, the density matrix is given as

Here, I represents the band index, k represents the electron wave vector, and 12, k) represents the Dirac state vectors to describe the electron states in the conduction and valence bands. The dimension of materials need not be of concern for the moment. The state vector satisfies the orthogonality relation Of (A‘, k’II, k) = 6;.;,6kk. and the complete relation of Cj.,k 12, k)(A, kl = 1. By taking the time differential of Eq. (1.17) and using the time-dependent Schrodinger equation, the density operator satisfies the equation of motion as aP(t)

-=

at

i --[H,+H,,p(t)] h

+(

-

(1.18)

where H , is the electron unperturbed Hamiltonian, and H r is the Hamiltonian representing the interaction between electrons and photons. The second term of the righthand side added phenomenologically represents relaxation due to the incoherent scattering processes. This relaxation is characterized by the longitudinal relaxation time constant for the diagonal density-matrix components and the transverse relaxation time constant for the nondiagonal density matrix components. Using the Dirac representation, the Hamiltonian for the kinetic energy of electrons is given as (1.19) The interaction Hamiltonian with the electromagnetic field is given as

where e is the electron charge, m, is the electron mass, and p is the momentum operator. A,&, t ) is the vector potential for the optical mode with the polarization unit vector, e,, the wave vector, q, and the frequency,

14

MlTSURU SUGAWARA

w. It is written in a classical manner as

A&,

t ) = fe,[Aq,u(w)e''~"-"') + C.C.]

(1.21)

The electric field is related to the vector potential as

Using the linear susceptibility, x ( l ) , and the third-order susceptibility, x(~), the polarization of the isotropic material is defined as P(r, t ) = Eofl)E(r, t )

+ E ~ X ( ~ t)E(r, ) E ( ~t)E(r, , t)

(1.23)

where E~ is the permittivity of a vacuum and subscripts of q and u are are omitted for simplicity. The expressions for the polarization, fl) and given by the density matrix that satisfies Eq. (1.18), as shown below. In the matrix element of Eq. (1.20), the k-vector selection rule between the conduction-band electron vector, k, , and the valence-band electron vector k,, holds as (see the appendix) k, z k,

=

k

(1.24)

Thus, the interband optical transition occurs perpendicularly in the energy band dispersion diagram, as shown in Fig. 1.3(b). Linear Susceptibility

The linear susceptibility and the linear optical absorption and gain spectra are derived here. Let us use a two-band model, that is, we consider the interband transition between one conduction band, denoted by c, and one valence band, denoted by v. The contributions from other bands can be added up later. One optical mode with a fixed wave vector, polarization, and frequency is taken into account because this is a situation in which we observe the linear optical absorption and gain of semiconductors. Thus, Eq. (1.19) becomes

where u&k is the frequency of the conduction-band electron with k, and wsk is the frequency of the valence-band electron with k. Equation (1.20) becomes H I = I*cr,k(t)lcrk)(v, kl + &,k(L)lv? k)
1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 15

with

where (1.28) (1.29)

and

See the appendix for the interband transition matrix element of P:L,.k.The density matrix of Eq. (1.17) becomes

The nondiagonal terms, pCDand pVc,are related to the polarization, and the diagonal terms, p,, and p,,, represent the population of the conduction and valence bands, respectively. From Eqs. (1.18), (1.25), (1.26), and (1.32), we get the motion of the nondiagonal and the diagonal terms as

and

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MITSURUSUGAWARA

Here, o,,,k = w,,k - uL..k; r,, is the transverse relaxation rate due to scattering; rTIis the longitudinal relaxation rate of the conduction-band electron; and r[ is the longitudinal relaxation rate of the valence-band electron. Due to intraband scattering. the relationship of jc,(k, t ) = -jJk, t ) does not always hold. These equations of motion can be solved by the conventional perturbation expansion of p = p(O) p ( l ) p(') . . . , where the number in parentheses represents how many times the electron has interacted with the electromagnetic field. The zero-th order term, p(O), represents the initial state of materials. Equation (1.18) can be separated according to the perturbation order. By expanding the density operator into the Fourier series of p(t) = X u ,p(o')e-iu'*,we obtain the first-order perturbation term as

+

+

+

(1.36) The diagonal terms, pi:) and p!:,', are zero. The linear polarization of the system is given by tracing the diagonal terms of the p ( ' ) ( o ) - polarization product as

(P:f:(o)/Z)

=

Tr[p(’)(o)P] = -

p$)(o)( jlee;rli)Qd i.j=c.r

where P is the polarization per unit volume in the e, direction, and the relationship of (1.38)

is used. Qd (d = 0,1,2,3) is the inverse of the crystal volume; Q3 = V-' for bulk materials with a volume of V ; Qz = ( D L Q W ) - ' for quantum wells with an area of D and a width of LQw;Q1 = N w i / L for quantum wires with an areal density of N w i and a length of L; and Qo = N , for quantum dots with a volume density of N , (see Fig. 1.2). The term p::) is usually replaced with the Fermi-Dirac distribution function of the valence band, f l , k , and pi:) with that of the conduction-band, J , k , as 1

+1

= exp[(hw,., - Pi)/kBT]

( i = c,v)

(1.39)

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 17

where p i is the quasi-Fermi level; k , is the Boltzmann's constant; and T is temperature. The value of p i is determined for 2Ckf;.k to give the total number of electrons for each band. The Fermi-Dirac distribution holds when electron-electron and electronphonon scattering toward thermal equilibrium occurs much faster than electron-optical interaction processes or other nonradiative recombination processes. This is not the case in semiconductor lasers operating at highoutput power, where the carrier-photon interaction is so strong that a large number of cavity photons extract carriers from the bands through the stimulated emission process with the cavity photon lifetime, and carrier relaxation cannot catch up with the extraction. This effect becomes remarkable especially in quantum-dot lasers, where the energy levels are discrete and the carrier relaxation lifetime is significantly slowed because of insufficient phonons to satisfy the energy conservation rule. This is known as the phonon bottleneck. The experimentally measured relaxation lifetimes are several tens to hundreds of picoseconds at a low excitation regime (see Chap. 5)-definitely lower than the typical cavity photon lifetime of ten picoseconds. Quantum-dot laser operation is simulated in Section V. From Eq. (1.23), the relationship between the Fourier components of the linear polarization and the electric field is

Thus, we get the first-order susceptibility as

(1.41)

Degeneracy due to spin is taken into account. The contribution of other k states and bands is included. Taking only the resonant term, the linear optical absorption coefficient, sc"'(w), or gain, g(’)(co), is given from the imaginary part of Eq. (1.41) as

18

MITSURUSUGAWARA

where ( 1.43)

is the normalized broadening function and nr is the background refractive index. The broadening obeys a Lorentzian distribution with a full width at half maximum (FWHM) of 2r,,, which generally increases with the number of carriers as a result of electron-electron scattering, and with temperature as a result of electron-phonon scattering. The refractive index, including the contribution of interband transition, is given from the real part as (1.44)

Let us perform the summation on the wave vector in Eq. (1.42) for the parabolic band with the dispersion of (1.45)

where E , is the band gap;

is the reduced effective masses; ELd) is the quantized energy in the conduction band; ELd’ is that of the valence band (see Fig. 1.3, depicting for quantum wells); and k(d) is the electron wave number in the free direction. By substituting the delta function for the broadening function of Eq. (1.43), we get an analytical expression for bulk (d = 3), quantum wells (d = 2), and quantum wires (d = 1) as (Haug and Koch, 1994)

x @(hw - E ,

-

E$)

-

(1.47)

Eid))[.f,(w) - f,(o)]

where L3 = I/; L2 = D; the integrations in the angular part are R, Q, = 2n, and Q, = 2; O(x)is the Heaviside step function; and (i

=

c,v)

= 471,

(1.48)

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 19

For quantum dots, by omitting the wave vector from Eq. (1.42), we get

where indexes c and v represent discrete energy states in the conduction and valence bands. By comparing the results in Section I1 with Eqs. (1.47) and (1.49), we see that the interband optical absorption spectrum directly reflects the density of states or, more correctly, the joint density of states between the conduction band and the valence band. The steplike absorption continuum in Fig. 1.4(a) in quantum wells fits well with Eq. (1.47) for d = 2, .f, = 0, and S, = 1. The sign of the absorption coefficient turns from positive to negative when f, - f, < 0; in other words, the optical gain appears. Nonlinear Susceptibility

The linear susceptibility obtained above describes the response of semiconductors when one electron interacts with one photon at one time. As the power of the electromagnetic field increases, higher-order effects become remarkable, where more than two photons participate in the electronphoton interaction within a coherence time. The way materials respond to the higher-order process is described by the nonlinear susceptibilities. This optical nonlinearity has a wide variety of applications, such as high-speed all-optical space and time domain switches, wavelength converters for wavelength division multiplexing systems, and conjugate wave generation for dispersion compensation. For the practical application of nonlinear devices, both large and broad-band nonlinear susceptibility is required. (Chapter 7 discusses the possibilities of quantum-dot optical nonlinear devices.) In semiconductor lasers, however, nonlinearity usually degrades laser performance, since the nonlinear optical gain causes gain saturation at a high cavity-photon number, leading to output-power saturation and limiting the highest modulation speed. Nonlinear susceptibilities can be derived by further continuing the perturbation expansion of the density-matrix equation. The two-band model used above describes the system in this case as well as long as the wavelengths of all the electromagnetic fields participating in the electron-photon interaction are close to the resonant wavelength of the two bands, that is, w ~ , ,In~ . this two-band model, even-order nonlinearity does not appear because the even-order interaction erases the polarization. To describe the second-order nonlinearity, a three-level system, as well as the crystal geometry without an inversion symmetry is required. The calculation of the second-order non-

20

MITSURUSUGAWARA

linear susceptibility is biased quantum wells with a tilted potential profile is presented by Kuwatsuka (1994). Let us derive the formulae for the third-order susceptibility. From Eqs. (1.33) through (1.35), the second-order components of the density matrix are

and p1.t;'(w) = 0

(1.52)

The second-order polarization vanishes, as seen from the nondiagonal term of Eq. (1.52). The third-order component of the nondiagonal term, taking only the resonant terms, is

The third-order polarization of the system is given by the diagonal trace as

(P$(w

=

+

2w2 - w J 2 ) = T r [ ~ ' ~ ) ( w )= P l-[pL~)(w)(ulreo.rlc) h.c.]Q,

(1.54)

and is related to the electromagnetic field as

( P g ( 0 = 2w,

-

cu1)/2)

= (&)%X(3Y2W2

- (111;

(023 w2,

-ol)E~,,,,(w,)Eqi.02(02)2cc/m21

(1.55) In the numerical coefficient in Eq. (1.55), 3! represents the order of the interaction and 2! indicates that two of the three electromagnetic fields

OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 21

1

cannot be discriminated. Thus, the third-order nonlinear susceptibility is given as

1 w2- w,,

+ ir,,

1

-

]

(ol - o,,~ - TCt,

[m2/V2]

(1.56)

Taking the light “2” with the frequency of w 2 as the pump beam, and the light “1” with w 1 as a probe beam, Eq. (1.55) describes the polarization to generate a signal beam with a frequency of w, = 2w2 - wl. If the two pump beams proceed in the opposite directions, the signal is the phase conjugate wave of the probe beam. See Chapter 7 for experiments on nondegenerate four-wave mixing for wavelength conversion and conjugate wave generation is dominated by the dephasin semiconductor lasers. The bandwidth of which determines the response speed of the ing rate of the polarization To,, nonlinear process (see Chapter 7 ) . Using f 3 ’ of Eq. (1.56) under the degenerate case of o2= ol, the optical gain, including the nonlinear effect, is given as

where the power density of the electromagnetic field is (1.58) Since ~ ‘ ~ ’ ( w has) the sign opposite to that of g(’)(w), the optical gain decreases as the power of the electromagnetic field in the cavity increases.

2.

SPONTANEOUS

EMISSION OF

PHOTONS

Let us consider a spontaneous emission process where an electron in the conduction band recombines with a hole in the valence band by emitting a photon. It is now well known that the time evolution, the spatial pattern,

22

MITSURUSUGAWARA

and the spectrum of the spontaneous emission depend on the electromagnetic environment of a material -that is, the optical modes -and can be modified by tailoring the modes using a cavity (for example, Yokoyama and Ujihara, 1995). The purpose here is to derive a formula for the spontaneous emission rate, or lifetime, in semiconductor nano-structures in the threedimensionally symmetric optical modes. The quantized expression for the electromagnetic field is used because the concept of spontaneous emission naturally comes out of the treatment. The remarkable difference from the derivation of the optical susceptibility in Section 111.1 is that all possible optical modes into which photons are emitted are taken into account. Using the quantized expression for the electromagnetic field, the vector potential of the quantized photons is written as

+

AqJr, t ) = A4,~(o,)e,[efq"h,(t)e-iq%,'(t)]

(1.59)

where

(1.60) 2: and h,, are respectively, the generation and annihilation operators of photons; v is the optical mode; oqis the frequency of the optical mode with the wave vector, q; and V , = L,3 is the quantization volume of photons. The quantization is done in the crystal as L = L,. To define the density of states for the electromagnetic fields, we use the periodic boundary conditions, where the wave function should be periodic in the x, y, and z directions with a period, L,. This gives q, = 2d/L,, q,, = 2nrn/L,, and q, = 2nn/L,, where I, m, and n are integers. Thus, the electron-photon interaction Hamiltonian for the interband transition is (1.61) Let us describe the temporal variation between the excited state with the electron in the conduction band and no photon Ic,k;O) and the ground state with the electron in the valence band and one emitted photon Iu, k; lq,u)using a time-dependent Schrodinger equation. The wave function for the system is written under the rotating wave approximation with only the resonant terms considered as

Let us assume the initial condition where one electron is in the conduction

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 23

band and the field is in the vacuum state with a(0) = 1 and b,,(O) = 0. The electron drops into the valence band by spontaneously emitting a photon with an energy of oqand a momentum of q. By substituting Eq. (1.62) into the time-dependent Schrodinger equation of [H,

+ HI]’€’ = ih-aaty

(1.63)

we get the following equations of motion for the probability amplitude as

and

where the coupling constant is ( 1.66)

The decay rate, y d , includes all the dissipation processes besides spontaneous emission, such as nonradiative recombination at crystal defects or impurities and scattering into other electronic states. Integrating Eq. (1.65) and then substituting it into Eq. (1.64), according to Weisskopf and Wigner (1930), we obtain b(t) =

-1Iyq.012f: q.0

k)(tpt’)l dt’ - ‘id a(t)

a([’)e[-i(cf)qpco
2

(1.67)

Let us transform the summation on q into the integration on the frequency, using the three-dimensional continuous optical mode density p(w). (Note that p ( o ) here is not a density matrix.) Then, Eq. (1.67) becomes

(1.68)

24

MITSURUSUGAWARA

where gw,ois the coupling constant at the frequency of o.This result shows that the coupling to the continuous optical mode always causes an exponential irreversible spontaneous emission. It can be visualized as an electron coupling to various optical modes with different frequencies and losing its memory as soon as it emits a photon due to the interference of the different optical modes. From Eq. (1.68), we get the spontaneous emission rate as

This is Fermi's golden rule. Let us calculate the spontaneous emission rate of Eq. (1.69) for quantum wells. As seen in Fig. 1.5(a), both the S-polarized mode (its electric vector normal to the plane defined by q and the z-axis) and the P-polarized mode (its electric vector in the plane defined by q and the z-axis) are taken into account. Thus, the coupling constant is given as

Z

4

FIG. 1.5. (a) S-polarized mode with its electric vector normal to the plane defined by q. and the z-axis and P-polarized mode with its electric vector in the plane defined by q and the z-axis. The UJ plane corresponds to the quantum-well plane. (b) Wave vector of the electromagnetic field, q = (q,,.4,). The xy plane corresponds to the quantum-well plane. The wave number of the electromagnetic field emitted by the interband transition is n,q,,k/c, where w , , , ~is the frequency dillerence between the k states of the conduction band and the valence band, In exciton emission, q = n,tu,/c, where w.,' IS the exciton energy.

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 25

where q = (qI1,qJ, q" = ( q x , 4), is the in-plane (quantum-well plane) component of the photon wave vector; q, is its perpendicular component; M is the matrix element between the conduction-band and valence-band edge states; Zncz,,t.Z is the overlap integral between the z-direction envelope wave functions for the electron and the hole quantized states; and M,, is the factor dependent on the electromagnetic field polarization direction of e l land e, and on the electron wave vector. Reflecting the nature of base fuctions, M,, differs between the electron-heavy-hole and electron-light-hole transitions (see the appendix). So, Eqs. (1.69) and (1.70) give

The next step is the summation on the possible optical modes that satisfy the energy conservation rule. Let us transfer the summation only on 9, into the integration, using the mode number of (LC/27c)dq,between q, + dqz and dq,. This is done to clarify the difference between the electron-hole transition and the exciton spontaneous emission treated in Section IV.5. Thus, we obtain

where

( I .74)

and q,

= n,w,,k/c.

Here, D

=

Lf

=

L2. The summation on the in-plane wave

26

M~TSURU SUGAWARA

vector is taken within qll < qc to satisfy the energy conservation rule (see Fig. lS(b)). Then, by integrating Eq. (1.72) into the quantum-well plane, we get

(1.75) The polarization dependence arising from the asymmetric characters of the base function of valence bands disappears from the final result because the optical mode distribution is symmetric. In Section IV.5, we will see how Eq. (1.75) is modified for the exciton spontaneous emission process. When there are N electrons in the conduction band and N holes in the valence band, the average spontaneous emission rate is given by summing Eq. (1.75) in k space as

The summation in k can be transformed into the integration in the energy via the same procedure used to derive the absorption coefficient in Eq. (1.47). Note that Eq. (1.75) holds not only in quantum wells but also in bulk materials, quantum wires, and quantum dots, simply by changing the matrix element. This is so because the three-dimensionally symmetric photon mode is considered and because the photon wave vector does not appear in the wave vector selection rule. For example, by omitting Zncz.,,zz in Eq. (1.75), the expression for bulk materials becomes ( 1.77)

Let us use material parameters of bulk In,~,,Ga,,,,As as me = 0.044m0; the spin-orbit splitting energy A = 0.35 eV; E, = 0.75 eV; M 2 = 2.12m,E, with D’ = 0 in Eq. (A.21), and n, = 3.5. Thus Eq. (1.77) gives rsD= 2.8 ns.

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 27

3. RATE EQUATIONS FOR LASEROPERATIONS Semiconductor lasers are the most important optoelectronic devices in modern optical technologies such as optical transmission systems, optical data storage, biomedical optoelectronics. The principle of laser operation is that light, first produced by spontaneous emission, going back and forth through the gain medium in a cavity is amplified as a result of the stimulated emission process, leading to high-power coherent light output if the gain exceeds the cavity loss. The optical gain appears in semiconductors when electrons and holes are injected into the conduction band and the valence band, respectively, to the extent that the absorption coefficient of Eq. (1.42) becomes negative. This situation is called populution inuersion, since in thermal equilibrium most electronic states in the conduction band are empty and those in the valence band are full. The semiconductor also acts as an optical wave guide to form a cavity using both side cleaved facets or, in the case of vertical cavity surface emitting laser diodes, distributed Bragg reflection mirrors. Laser performance has improved steadily over nearly three decades since room-temperature continuous operation of double heterostructure lasers was first achieved (Hayashi et al., 1970), leading to high-performance strained quantum-well lasers (Thijs et al., 1990). See Chapter 6 for an overview of recent developments in quantum-dot lasers and their significance and future prospects. Laser operation is defined by a set of rate equations to describe the time-dependent variation of carriers and cavity photons. Using rate equations, various laser properties such as lasing threshold currents, threshold carrier density, current-power relationship, external differential quantum efficiency, and small-signal modulation bandwidth can be simulated. In this section, we summarize the rate equations that will be used in the discussion of quantum-dot lasers in Section V.l. The most common laser is an edge-emitting type with cleaved facets (Fig. 1.6(a)). Currents are injected into the active region through the p-n junction, and the optical wave guide confines the light to the active region's vicinity. The smooth cleaved facets with the reflectivity of R , and R , act as both a reflector and an output coupler with the back facet usually coated to increase the reflectivity. The typical dimensions are a cavity length, L,,, of between 200 and 1500 pm and a lateral width of 2- 10 pm. Several requirements must be satisfied by the materials, such as band gap, refractive index, and lattice constant. The rate equations to describe the time-dependent variation of the active region carrier number, N , and the cavity photon number, S, are

(1.78)

28

MITSURUSUGAWARA

FIG. 1.6. (a) Cavity laser. (b) Schematic plot output power and carrier number versus injected current in a cavity laser. Above the threshold of If,,.the injected current is transformed into photons, keeping the carrier number clumped. The nonlinear gain causes gain saturation at high output power, leading to a decrease in the quantum efficiency and an increase in the carrier number.

and (1.79)

where I is the injected current; r is the optical confinement factor; C , is the spontaneous emission factor representing the fraction of spontaneous emission entering the lasing mode; E, is the nonlinear gain coefficient derived from Eq. (1.57); and V , is the active-layer volume. (See Section V.1 for an explanation of 8, in quantum-dot layers.) The photon lifetime in the cavity, z p , is written as

where ai is the internal loss. Carriers injected into the active region recombine radiatively with the lifetime of rsp or nonradiatively with the lifetime of z,, giving the total lifetime as

In long-wavelength infrared lasers, nonradiative recombination includes Auger as well as impurity-related recombination. The optical gain for the lasing mode, g:), is usually the maximum value of g(l)(a) in conventional

1

OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 29

Fabry-Perot lasers, since the separation of the cavity mode is usually less than 1 nm. The output power is proportional to the cavity-photon number and is given as Po,,

=

hwcS In(1/R)/(2Lcanr)

(1.82)

where R is R , or R , and Aw is the emitted photon energy. The temporal variation of S and N , including the rise time of the response, the relaxation oscillation, and steady-state values, etc., can be obtained by numerically solving Eqs. (1.78) and (1.79). The steady-state solution of the rate equations is given by setting the lefthand time-derivative term at zero in Eqs. (1.78) and (1.79). For Eq. (1.79) to have a solution with S > 0, the lasing threshold condition is given as T; = (c/n,)Tg!& with the threshold gain of g:,',,, and thus,

The contribution of C, is disregarded for a conventional cavity laser. Let us check the solution simply by setting an approximate formula for the optical gain as

where ga)' is the differential optical gain and N , , is the carrier number for the transparency where the gain is zero. This formula is often used to express the optical gain in bulk semiconductor lasers. For quantum-well lasers, an empirical formula using a logarithmic dependence of carrier is usually employed (Chuang, 1993). The solutions are N

= z,I/e

(1.85)

and

s=o

(1.86)

below the lasing threshold, and (1.87)

30

MITSURUSUGAWARA

and S

= (t,/e)(f - eN/z,)

(1.88)

above the threshold. The threshold current is given from Eqs. (1.87) and (1.88) with S = 0 as (1.89) Figure 1.6(b) is a schematic graph of the output power and carrier numbers as functions of the injected current. The carrier number is clumped above the threshold, since the injected electrons are transformed to photons through stimulated emission. An increase in the injection current above the threshold of Eq. (1.89) leads to an almost linear increase in output laser power until its saturation occurs. The saturation of power and the increase in carrier number are due to the nonlinear gain saturation effect through the third-order nonlinear gain coefficient. A procedure to design strained quantum-well lasers with low-threshold currents is as follows. First, the band dispersion curves for the conduction and valence bands are calculated by means of the k * p perturbation method, the curves are quite sensitive to the amount of strain and the well width. and the optical gain are Then the spontaneous emission lifetime (z), calculated as a function of carrier density through Eqs. (1.42) and (1.76). Next, the strain and well-width dependence of the threshold currents are obtained. In strained InGaAs/InP quantum-well lasers, it has been shown that either direction of strain -that is, bi-axial compressive or tensile lowers the threshold currents (Sugawara and Yamazaki, 1994). In this simple description of laser operation, it is assumed that all of the carriers are injected into the active region without a time delay. This is not a good assumption, especially in quantum-dot lasers, since the carrier relaxation is retarded due to the complete discrete level formed in quantum dots. The laser simulation including retardation of carrier relaxation is presented in Section V.l.

IV. Exciton Optical Properties In Section 111, optical susceptibility (absorption and gain) and spontaneous emission due to interband electron- hole transition were discussed. The absorption spectra have the same energy dependence as that of the interband joint density of states, and they vary with the dimensions of the

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 31

nano-structures. As seen in the experimental optical absorption spectrum of Fig. 1.4(a), however, strong resonant peaky spectra are superimposed on the steplike absorption continuum peculiar to quantum wells. This is due to the creation of electon-hole bound states, called excitons, through the Coulomb interaction. Since the discovery of strong exciton resonance in the absorption spectra (Dingle, 1975), excitons in quantum wells have revealed various unique properties, which are not only physically attractive but also useful for optical device applications. First, exciton resonance is observed even at room temperature, owing to a variety of factors: the enhanced oscillator strength and binding energy caused by wave-function compression under the potential barriers; the relatively low scattering rate (several hundred femtoseconds at room temperature, as will be seen in the experiments depicted in Fig. 1.9 on exciton spectrum broadening); and high-quality crystals with smooth interfaces and homogeneous compositions. Second, remarkable optical nonlinear effects have been found under both real and virtual excitation. In particular, the a.c. Stark effect, which shows a blue shift of exciton resonance and its bleaching under virtual excitation, has attracted much attention as a possible mechanism for sub-picosecond switching and as one of the most fundamental phenomena in light-matter coupling (Schmitt-Rink et al., 1989, for a review). Third, when the electric field is applied perpendicularly to the quantumwell layers, exciton resonance exhibits an electric-field-induced shift toward the lower energy region (Chemla et al., 1983). What is unique is that the resonance clearly remains up to an electric field of about lo5V/cm because the potential barriers prevent the ionization of excitons. This so-called quantum-confined Stark effect shows a significant change in the absorption coefficient and is now being used in high-speed optical modulators up to 10 GHz. The bistable devices- SEEDS-operate on the basis of this phenomenon (Miller et al., 1985b). Fourth, spontaneous emission of excitons in semiconductor nano-structures has led to various interesting phenomena, such as rapid spontaneous emission on the order of picoseconds in quantum wells (Hanamura, 1988), mesoscopic enhancement of the spontaneous emission rate in microcrystals (Hanamura, 1988), and the cavity-polariton- that is, the exciton-photoncoupled mode- in semiconductor microcavities (Houdre et al., 1994, Cao et al., 1995). These phenomena come from a unique selection rule to preserve the wave vectors between the exciton center-of-mass motion and the photon. Fifth, the possibility of exciton lasing in wide-gap semiconductor materials such as the II-VI and GaN systems, which has been frequently discussed in recent years (Ding et al., 1992, 1993). We should discount the simple

32

MITSURU SUGAWARA

explanation that the high exciton-binding energy of these wide-gap materials stabilizes exciton states, resulting in exciton lasing. Excitons as bound states between an electron and a hole cannot exist under the population invertion (Uenoyama, 1995). Thus, in order for bound-state excitons to generate a gain, the exciton emission energy should be shifted from the resonant exciton absorption energy through some mechanism, such as scattering (Galbraith, 1995, for example), exciton localization, and bi-exciton formation (Sugawara, 1997a, 1998), etc. This section reviews the optical properties of excitons. In Section IV.l through 3, using the second quantization, formulae are introduced for exciton state vectors, effective-mass equations, and the interaction Hamiltonian with photons. In Section IV.4, a formulae for the optical absorption spectra is derived and compared with the experiments. In Section IV.5, the spontaneous emission of excitons is discussed. Also, a quantum disk is treated to show how the emission properties vary from quantum wells to quantum dots by a change in the disk radius. In Section IV.6, the disk is placed in a planar microcavity so that the spontaneous emission behaviors of cavity-polaritons can be observed. In Section IV.7, the effect of the electron-hole Coulomb interaction on optical gain is briefly reviewed. For details on the quantum-confined Stark effect, refer to Chuang’s textbook (1995).

1. STATEVECTORS An exciton is composed of an electron in the conduction band and a hole in the valence band under the Coulomb interaction. A hole is a positive particle representing an empty state in the valence band. Its effective mass is defined as mf = -mz, and its sign is positive in most cases, as stated before. The hole energy is at its minimum at the top of the valence band and increases along the valence-band dispersion curve. The creation operator of a hole with a wave vector in the valence band, k, is defined as (1.90) where ak,, is the electron annihilation operator in the valence band, and

k h --ku -

(1.91)

The conduction-band effective mass and the wave vector are renamed for the electrons as m,*and k,. The position vectors of an electron and a hole are re and r,, respectively.

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 33

The wave function of the exciton state-an electron-hole pair state-is given by the electron creation operator, a:, and the hole creation operator, b;, on the ground state (filled valence band and empty conduction band) and by summing the pair function over all possible states as (1.92)

To determine the expansion coefficient, Ck,kh, in Eq. (1.92), let us replace the creation operators of an electron and a hole with the field operator, using the relationship below (Haken, 1973, for example):

(1.93)

and

Here, Fk,(re) and Fkh(rh)are solutions to a one-particle effective-mass equation for an electron and a hole, respectively. By substituting Eqs. (1.93) and (1.94) into Eq. (1.92), we get r r

1

(1.95) Here, t,hex(re,rh) satisfies the effective-mass equation for excitons, as will be shown below. The exciton state laex)is given by the overlap of the created electron-hole pair with the probability amplitude of +ex(re, rh).The exciton state is described in the wave vector space by expanding the field operator in Eq. (1.95) as

(1.96)

34

MITSURUSUGAWARA

and

Then, by substituting Eqs. (1.96) and (1.97) into Eq. (1.95), we get the exciton state vector as

(1.98) where A(k,,k,) is the Fourier transform of the exciton envelope wave function in real space. When the Coulomb interaction is neglected and the envelope wave function is given by the product of the electron plane-wave with k, and the hole plane-wave with k,, Eq. (1.98) becomes

Comparing Eqs. (1.98) and (1.99), it is clear that the Coulomb interaction makes up the exciton state by summing the electron and hole pair state with the amplitude of A(k,, k,) around the band edge. The Coulomb potential between an electron and a hole is given by (1.100)

where E, is the static dielectric constant. The potential depends only on the difference between the electron and hole position vectors. Therefore, let us change the coordinate vectors into the relative motion vector as r = re - rh

(1.101)

and the center-of-mass motion vector as

R=

+

m$re mtrh M*

(1.102)

where M*

= m:

+ mt

(1.103)

1

OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 35

Transforming the wave vectors as

K

= k,

+ k,,

(1.104)

and (1.105) we get

A(k,, kh) = A(K, keX)= V

d3rd3R$,,(r, R)e-'K.Re-'kex'r( 1.106)

The above equations can be applied to exciton states regardless of the dimension of materials. However, it is more convenient to use the quantizedstate wave function explicitly in expanding the field operator. This is done for both quantum wells and quantum dots below.

Quantum Wells When the quantum-well confinement potential in the z direction is much greater than the electron-hole Coulomb potential energy, the z-direction wave function can be separated from the wave function for the in-plane motion. Thus, the field operator is expanded as (1.107)

and

where ri = (r\I,zi);ki = (k\I, niJ; and i and (1.108), we get

= e, h.

Then, from Eqs. (1.95), (1.107),

(1.109)

36

MITSURUSUCAWARA

where $ex(rll, R") is the in-plane envelope wave function separated from the z-direction component. By transforming the variables, the Fourier component is written as

The vectors related to the relative motion and center of motion are all two-dimensional. Quantum Dots Let us expand the field operator as (1.111)

and

where k , given as

= (nex,ney,nez)

and kh = (nhx,nhy,nhz). Thus, the state vector is

The integration in Eq. (1.1 13) represents the mixing of different quantized states via the Coulomb interaction. Without the Coulomb interaction, Eq. (1.1 13) is reduced to Eq. (1.99).

2. EFFECTIVE-MASS EQUATIONS The effective-mass equation for the electron-hole system is given as

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 37

Here, the Hamiltonian is as follows (Haken, 1973, for example):

where H ; is the electron kinetic energy; H i is the hole kinetic energy; V(r, r’) is the Coulomb potential energy of Eq. (1,100) between two particles; and the field operator working on both the conduction and valence bands is

By substituting Eq. (1.116) into Eq. (1.1 19, the Hamiltonian becomes

where

for the electron, with the energy origin taken at the valence-band top, (1.119) for the hole, and

( 1.120)

38

MITSURUSUGAWARA

The commutation relation of the field operators are used (Haken, 1973, for example). V,, is the electron-electron interactions; V,, is the hole-hole interaction; V,, is the electron-hole interaction; and V,, is the exchange interaction. By substituting the exciton state vector of Eq. (1.95) into the effectivemass equation of Eq. (1.1 14), we get the exciton effective-mass equation as

where the exchange interaction term is neglected. Analytical solution to Eq. (1.121) are derived for both the three-dimensional (Elliot, 1957) and the two-dimensional cases (Shinada and Sugano, 1966). In Chuang’s textbook (1995), the results are instructively summarized. Equation (1.121) has bound-state solutions for E,, < E , with a series of discrete energies, and unbound-state solutions for E,, > E , with continuum energies. In bulk materials, the bound-state energy is given as E,,

= E, -

m:e4 32rc2h2&;n2

2K2 +-k2M*

(1.122)

where n = 1,2,3. .. . The wave function is separated as (1.123)

where the ground state for the relative motion (n

4(r)= rc -112 aB-312 with the Bohr radius a, center-of-mass motion is

=

exp(-r/aB)

= 4ne0~,h2/(m~e2), and

1) is (1.124)

the wave function for the

(1.125)

Here, several other cases are briefly described: quantum wells, quantum wells with in-plane confinement potential; and quantum wells under a magnetic field.

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 39

Quantum Wells It is assumed that quantum wells are type I; that is, both an electron and a hole are confined in the same well layer. Though the asymmetry of the effective mass is taken into account for the direction perpendicular to and parallel to the well layer, band nonparabolicities are neglected. The superscripts of 11 and I are attached to the effective masses and position vectors to describe the parallel and perpendicular directions, respectively. By adding the quantum-well confinement potential to Eq. (1.121), an effective-mass equation results:

where peh = [r"' + (z, - z , ) ~ ] " ~is the distance between an electron and a hole, and ((ze) and V,(z,) are the z-direction quantum-well confinement potentials for an electron and a hole, respectively. When the confinement potential in the z direction is much greater than Coulomb potential energy, the wave function can be separated as

This is a good approximation for actual quantum wells with several tens to hundreds meV potential barriers. The in-plane relative motion and the center-of-mass motion can be separated. The exciton resonance energy is given as (1.128)

where Enpl and En,,=are the electron and hole quantized energies in the quantum well. The energy of ELm for the relative motion is determined by ( 1.129)

and

40

MlTSURU

SUGAWARA

+

where n = 1,2,3,. . . and rn = 0, 1,. . . , +(n - 1). The optically active exitons are s excitons with rn = 0. For the ground-state optically active 1s excitons (n = 1 and rn = 0), a variational wave function of (1.131) is often used. The variational parameter, ,Iex represents the two-dimensional = ( ~ ( r ~ ~ ) ~ r lThe l ~ ~energy ( r ~ ~of)E) R, for the centerexciton radius, since ,Iex of-mass motion is ( 1.132)

and the wave function is (1.133) The in-plane wave vector is given by the periodic boundary condition as KII = 27c(n,/L, n,/L), where D = Lz and n, are integers (i = x and y).

Quantum Wells with In-Plane Harmonic-Oscillator-Type Parabolic Potential Since actual quantum wells inevitably have local structural imperfections, such as interface roughness and composition fluctuations, excitons are localized at the potential minima caused by the imperfections, especially at low temperatures. Their extreme cases are the quantum dots in Fig. l.l(b), where the potential depth is large enough to prevent exciton thermal escape even at room temperature. By adding the in-plane confinement effect to the quantum-well potential, we get the potential as

O(x) is the Heaviside step function (O(x) = 1 for

x 3 0 and O(x) = 0 for

x < 0). If the z-direction confinement potential is high enough to confine

excitons in the in-plane two-dimensional regions, Eq. (1.134) is reduced to approximately

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 41

These are exact when the z-direction potential height is infinite. Thus, the wave function can be separated as

using the z-direction quantized electron and hole wave functions. The exciton resonance energy is given as E,, = Eg Enez En,,= E,. The binding energy E , is determined by

+

= EB$e,(rll, RII)

+

+

( 1.137)

Equation (1.137) can be separated into the relative-motion and centerof-motion parts by using the following harmonic-oscillator-type parabolic potentials to describe lateral confinement for both an electron and a hole:

Then,

and (1.140)

For relative motion, we obtain

where $, is given by Eq. (1.130). For the ground-state s-state optically active excitons (n = 1 and rn = 01, we solve Eq. (1.141) by a variational method using the trial wave function of

where a', b', and c' are variational parameters, two of which are independent because of the normalizing condition. For center-of-mass motion, the

42

MITSURUSUCAWARA

effective-mass equation is

Equation (1.143) describes a two-dimensional harmonic oscillator and can be solved analytically. The wave function is given as

(1.144) where

p = ,,hRII/B,

nomials; k

B

d m , and

= = 0,1,2,3,.. . ; and

E;,

I

= 0,

= (2k

Llfl(p) are associated Laguerre poly? 1 , . . . , k. The energy is given by

-

111

+ l)tiu,

(1.146)

The ground-state wave function ( k = I = 0) is the Gaussian function.

Quantum Wells Under the Mugnetic Field Let us consider a situation where a magnetic field is applied perpendicular to the layer (z direction): B = (O,O, B). Taking the vector potential that works on an electron and a hole as ( i = e and h)

the effective-mass Hamiltonian for excitons is written as

where p, and ph are the momentum operators. First, we perform a transformation to the center-of-mass motion and relative motion coordinates.

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 43

Second, we choose the envelope wave function as (Knox, 1963)

where Ar(rll) = ( - yB/2, xB/2). Thus, we obtain

(1.150) with the exciton resonance energy of

(1.151) The expectation value of the third term (the Zeeman term) and the fifth term of Eq. (1.150) is zero for optically active S-state excitons. The fourth term, the diamagnetic energy term, dominates the magnetic-field-induced shifts of exciton resonance in quantum wells.

3. EXCITON-PHOTON INTERACTIONS The Hamiltonian describing the interaction between the electromagnetic field and the material is written in the second quantization expression as

(1.152) The vector potential of the quantized photons is given by Eq. (1.59). By substituting the field operator of Eq. (1.116) into Eq. (1.152), and selecting only the terms for the interband optical transition between the conduction band and the valence band, we get

44

Mirsuuu SUCAWARA

By expanding the field operator as (1.154)

and (1.155)

Eq. (1.153) becomes

+

The wave vector selection rule is q = k, ke and is already applied for the Note that the band-edge Bloch functions interband matrix element of PgSk. of u,(r) for the conduction band and ul,(r) for the valence band are added in Eqs. (1.154) and (1.155), but were dropped in the effective-mass equations. Let us derive the exciton-photon interaction Hamiltonian under the Dirac representation. The exciton state with n photons is

and the ground state with n

+ 1 photons is Is> = I q J ~ + 1)

(1.15 8 )

Coupled modes between excitons and photons are not taken into account here. The Dirac representation for the optical transition between the exciton state and the ground state is given as

where, using Eqs. (1.98) and (1.106)

(1.160)

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 45

and

In Eq. (1.161), the relationship of ~k,,e-'k'x'r = V6(r) is used. Care must be taken not to confuse the Fourier transform of the exciton wave function and the vector potential of the electomagnetic field. The band-edge value of Pzv,k is used for excitons, since exciton states are composed of the k-states around the band edge. The wave vector selection rule for the center-of-mass motion is K = q, and so the excitons emit photons in the direction of K. This concept is a key to understanding the exciton-polariton and fast spontaneous emission in quantum wells. The above equations can be applied to exciton states, regardless of the dimension of materials. However, it is more convenient to use quantizedstate wave functions explicitly in expanding the field operator. This is done in the following equations for both quantum wells and quantum dots: Quantum W e h

By expanding the field operator as (1.162) and

(1.163)

Eq. (1.153) becomes

In the Dirac representation of Eq. (1.159),

(1.165)

46

MITSURUSUGAWARA

and

Quuntum Dots

By expanding the field operator as (1.167) and (1.168) Eq. (1.153) becomes

H,

e

=mO

C 1 Aq,(~~)[P",l,a,a~b,',h + h.~.]

(1.169)

q,o k,.kh

In the Dirac representation of Eq. (1.159) (1.170) and

If the general formulae of Eqs. (1.98) and (1.06) are used, P:;

P&A(q,kex) % P&

= kex

s

d3R$,,(0, R)e-'q'R2 P;"

s

d3R$,,(0, R)

(1.172)

The last approximation holds when the dot volume is smaller than the exciton resonant wavelength and the electric dipole approximation holds that is, eciq'Rz 1. Under this approximation, the square of the matrix

1

OPTICAL PROPERTIES OF SEMICONDUCTOR

QUANTUM NANO-STRUCTURES 47

element is proportional to the volume covered by the center-of-mass motion. As will be seen in Section IV.4, the spontaneous emission rate of excitons increases in proportion to the dot volume. This is often referred to as the mesoscopic enhancement of the exciton spontaneous emission rate. In contrast, the spontaneous emission rate of the electron-hole pair is independent of the crystal volume and dimension, as seen in Section 111.2.

ABSORPTION SPECTRA 4. OPTICAL Let us use a density-matrix equation to derive the formula for the optical absorption caused by excitons. The procedure is almost same as that used in deriving linear susceptibility without the Coulomb interaction in Section 111.1. Slight differences are the base functions and the initial condition. The base functions are taken as the exciton state, le), and the ground state, Is), where le> = laex) and

Is)

=

Pg)

(1.173)

The system is initially in the ground state. Note that, as the number of excitons increases, the system is transferred to the electron-hole plasma state, making it impossible to use the exciton base functions. The classical expression for the electromagnetic field of Eq. (1.21) is used. The kinetic energy term is H,

+

= ~cJ~ele)(el ho,Ig)(gl

(1.174)

The perturbation term for the exciton-photon interaction is

with

+

,ueg(t) = +[peg(o)e-iw' peg(-o)e'"']

(1.176)

where (1.177) and (1.178)

48

MITSURUSUGAWARA

The density matrix for the exciton state is

The initial condition is p::) = 0 and p::) = 1. By solving the density-matrix equation as in Section 111.1, we obtain the nondiagonal term under the first-order perturbation as (1.180)

where we, = oj, - 01,. diagonal trace as

The polarization of the system is given by the

Degeneracy due to spin is taken into account and the relationship of (1.182)

is used. The summation over the exciton states is also taken into account. From its imaginary part, the optical absorption coefficient is given as (1.18 3)

Comparing Eq. (1.142) and Eq. (1.183), the exciton effect on the optical absorption spectra appears in the transition matrix of P:; and the transition energy of a,,.These are determined by the exciton effective-mass equations, The optical absorption spectrum consists of a series of resonance, the continuum spectrum whose intensity is affected by the Coulomb interaction, and the smoothly connecting part in between. The enhancement ratio of the continuum spectrum strength and the strength of the noninteracting electron-hole pair is known as the Somrnerfeld factor. The Coulomb effect on the optical absorption spectra is schematically summarized for one- two, and three-dimensional cases by Ogawa (1995). For quantum wells, using Qd = (DLQw)-', Eq. (1.1 85) becomes

ELQW= SP*BPl(hC1)- hoe,)

(1.184)

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 49

with the integrated intensity of (1.185) the oscillator strength per unit area of ( 1.186)

and the spectrum-broadening function of

B,(hw - hw,,)

=

hLg/n ( h -~

+

(1.1 87)

When probing light is applied perpendicularly to the quantum-well layer, ql1= 0, and thus KII = 0 from the wave vector selection rule (see Eq. (1.166). The exciton optical absorption strength is proportional to Id(rl1= 0)l2, which represents the probability of finding the electron and the hole at the same position. For the ground-state excitons in quantum wells, Ig5(rl1= 0)l2 is inversely proportional to the square of the two-dimensional exciton radius-that is, to as seen in Eq. (1.131). Equation (1.187) shows that the exciton optical absorption spectrum has a Lorentzian profile, with an FWHM of 2hTeg, which increases with an increase in temperature due to scattering by phonons. Figure 1.7 shows the calculation of the ground-state exciton resonance characteristics for various 111-V and 11-VI semiconductor quantum wells (b) binding (Sugawara, 1992): the well-width dependence of (a) radius, ,Iex; energy, E,; calculated by Eq. (1.126); and (c) integrated intensity, Sex, calculated by Eq. (1.185). The exciton radius decreases as the well width decreases due to the quantum-confinement effect, and increases in wells narrower than 2 nm due to the breakdown of the confinement effect -that is, the spread of the z-direction quantized-state wave function to the barriers. In conjunction with this, the binding energy increases as the well width decreases to about 2nm, and then goes down. The variation in the integrated intensity shows almost the same tendency as the variation in the binding energy. The decrease in narrow quantum wells is due to both the increase in the radius and the decrease in the electron-hole overlapping integral. Figure 1.7(d) shows the band-gap dependence of the integrated intensity in 5-nm quantum wells. The dashed line serves as a visual guide. Wider-gap semiconductors have a smaller radius, a greater binding energy,

50

M ITSURu SUGAWARA

'ZnMnSsRnSe

10 (a)

-

210

20

,

,

,

,

s

,

,

,

.

r

2

10

"

"

"

"

'

8

s!

s ! . ZnSaRnMnSe

ZnSelZnMn%sQ

z

6 -

.1:tn 5 4-

CdZnTmTm

.-

C

c '0 C 0

cn

Y

InGaAsilnP

C

/

9'

CI

C

20

Well width (nm)

(b)

.

"

I

0

X Y

O;'

Well width (nm)

C'

I

J&aPIIIGaInP-

.

2.

*.g

d""

GaAa(AIGaAs . d%GaAallnP

FIG. 1.7. Characteristics of 1s exciton resonance calculated for various 111-V and 11-VI semiconductor quantum wells (from Sugawara. 1992). (a) Well-width dependence of the radius, &. (b) Well-width dependence of the binding energy, E,. (c) Well-width dependence of the integrated intensity of the optical absorption spectra. (d) Band-gap dependence of the integrated intensity in 5-nm quantum wells. The dashed line serves as a visual guide.

and stronger resonance. This is due to their larger effective masses and smaller dielectric constants.

Experiment on Exciton Optical Absorption Intensity Though the quantum wells used in the calculation in Fig. 1.7 have actually been grown, so far there are no systematic experimental analyses of

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 51

how the exciton resonance intensity depends on materials. Here, instead of changing materials, the exciton in-plane wave function is experimentally compressed under a magnetic field perpendicular to the quantum-well layers to observe how the exciton resonance intensity depends on the exciton radius. Figure 1.8(a) shows the magneto-optical absorption spectra for the Ino.6sGao,,sAS/InP quantum well, whose cross-sectional TEM photograph is shown in Fig. l.l(a). The ground-state electron-heavy-hole (le-hh) exciton resonance at the absorption edge shows diamagnetic shifts, and its intensity increases considerably. The absorption continuum separates into discrete spectra, which are assigned to 2s states of le-hh excitons. Figure 1.8(b) shows the diamagnetic shifts as a function of the square of the magnetic field. The results for another quantum well with a composition of In,,,,Ga,,,,As/InP are also plotted. The curves are fitted by Eq. (1.150) for the S-state optically active excitons using a variational wave function for the relative motion of Eq. (1.142). The third and the fifth terms are zero. Taking into account the nonparabolicity of the conduction and valence bands, the energy is calculated in the wave vector space using the Fouriertransformed exciton wave function. The conduction-band dispersion is calculated by the first-order 8 x 8 matrix (Sugawara et al., 1993a), and the valence-band dispersion is calculated by the Luttinger-Kohn 6 x 6 k * p matrix (Luttinger and Kohn, 1955). The fitting parameters are valence-band Luttinger-Kohn parameters. The band-edge masses are rnL1 = 0.047rnOand mil = O.lrn, for In,,6,Ga,,,,As/InP quantum wells, and rn:’ = 0.05rn0 and m)ll = 0.12m0 for 1n0,,,Ga,~,,As/1nP quantum wells. If the magnetic field is treated as a perturbation for exciton states in the low-field limit and uses the hydrogenic wave functions of Eq. (1.131), Eq. (1.150) gives a diamagnetic shift of

AEr = 3e2&B2/16rn,*II

( 1.188)

which is proportional to the square of the magnetic field (dashed line). The approximation of the low-field limit holds at most up to 1 or 2 Tesla. Under higher magnetic fields, diamagnetic shifts are no longer proportional to the square of the magnetic field and deviate from the straight line, indicating that the in-plane exciton wave function shrinks (Aex decreases) under the magnetic field. The integrated intensity of the le-hh exciton resonance is plotted as a function of the magnetic field in Fig. 1.8(c). Here, the curves are the calculation of Eq. (1.185). Actually, the calculation was done in wave vector space and band nonparabolicities were taken into account. The secondorder k * p term in Eq. (A. 21) is set at D' = - 6 as a fitting parameter, giving

52

MITSURU SUGAWARA

I'

6

4l 8

3- ----4.2Tesla

' \

-7.OTesla

--.x=0.35,

(a)

'

.\

'

.-

-

'

In,pa,AdlnP IQWS on (001) InP I

Wavelength ( p)

0.5' 0

(c)

"

2

Lz=9.7nm

Magnetic field (T*)

. 4

.

'

6

.

1 8

Magnetic field (Tesla)

FIG. 1.8. Experiments on exciton diamagnetic shifts (from Sugawara, 1993a. Copyright 1993 by The American Physical Society). (a) Magnetooptical absorption spectra of the In,~,,Ga,,,,As/lnP quantum well, shown in Fig. l.l(a). (b) Diamagnetic shifts of the le-hh exciton resonance as a function of the square of the magnetic field. The results for another are also plotted. The curves are fitted quantum well, with a composition of ln,,,,Ga,,,,As/InP, by Eq. (1.150) for the IS-state optically active excitons using the variational wave function of Eq. (1.142). The dashed straight line is for the Iow-field limit by Eq. (1.1823). The downward bending of the diamagnetic shifts shows that the in-plane exciton wave function shrinks (,?e.r decreases) under the magnetic field. (c) Integrated intensity of the le-hh exciton resonance as a function of the magnetic field. The curves are the calculation of Eq. (l.lX5). The increase in intensity is due to exciton shrinkage.

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 53

about a 1.2-times larger value of M 2 than that given by neglecting D’ in Eq. (A.21). The excellent agreement between the calculation and the experiments shows that the enhancement of integrated intensity is due to the shrinkage of the exciton wave function, and it proves the derived theoretical expressions for the exciton optical absorption strength. Experiment on the Resonance Spectrum ProJile

In addition to the broadening due to scattering by Eq. (1.187), in actual quantum wells we must take into account inhomogeneous broadening due to spatial fluctuations of the resonance energy. The excition resonance energy depends on the structural imperfections of sites in quantum wells, and each level shows thermal broadening caused by scattering. Therefore, the spectrum-broadening function should be written using the convolution integral as B,,(hw - EeX)=

BO(ho - E,,

+ E)BT(E)d E

(1.189)

where B , is the inhomogeneous broadening function. It is assumed that the exciton scattering lifetime is independent of the slight variation in exciton energy. This is reasonable, since structural imperfections primarily change the band-gap energy or quantum-confinement energy, and the change in the exciton-binding energy is negligible. Figure 1.9 shows the measured (solid lines) and calculated (dashed lines) optical-absorption spectra at (a) 4.2 K and at (b) 295 K of In,,,,Ga,,,,As/ InP quantum wells used in the magneto-absorption measurements of Fig. 1.8. At 4.2 K, the electron-heavy-hole exciton resonance spectrum has a Gaussian distribution of

with an FWHM of To = 2.35(, = 4.7 meV. The calculated curve at 295 K fits the profile of the resonance very well, using Eq. (1.189) with re;' = 150 fs. Figure 1.9(c) shows the FWHM of the electron-heavy-hole exciton resonance spectrum as a function of temperature for different samples with almost the same composition and well width (10-nm Ino~,,Ga,,,,As/InP quantum wells) but with different inhomogeneous broadening. The solid

54

MITSURUSUGAWARA

"

1.40

(a)

1.&

1.!ill

Wavelength (pn)

Wavelength (pm)

(b)

2 0 h . I

-

, . ,

.

A

E

2

2.

i

u)

u

OO

(c)

100

200

300

400

Temperature (K)

FIG. 1.9. Experiments on exciton optical absorption spectra (from Sugawara et al., 1990. Copyright 1990 by The American Physical Society). (a) Measured (solid line) optical absorption spectrum of In,,,,Ga,~,,As/InP quantum wells at 4.2 K. The dashed line is Eq. (1.190) with an FWHM of 4.7meV. (b) Measured (solid line) optical absorption spectrum of In,,,,Ga,,,,As/lnP quantum wells at 295 K. The dashed line is calculated by Eq. (1.189) with r,' = 150 fs. (c) FWHM of the ground-state electron-heavy-hole exciton resonance spectrum as a function of temperature for three samples with almost the same composition and well width (10-nm In,~,,Gao~,,As/lnP quantum wells). The solid curves are calculated by Eq. (1.189),assuming the temperature-dependent phonon scattering rate of Eq. (1.191). The dashed line is the thermal broadening due to Eq. (1.187).

curves are calculated assuming the temperature-dependent phonon scattering rate of

reg= rph[exp(ho,,/kT)

-

13-l

(1.191)

with the LO phonon energy of hw,, = 30 meV and r;,,' = 65 fs. The dashed line is the FWHM of Eq. (1.187). Calculated values for the three samples are in good agreement with measurements, supporting the assumption

1

OPTICAL PROPERTIES OF SEMICONDUCTOR

QUANTUM NANO-STRUCTURES55

concerning the origin of scattering. Since the LO phonon energy of 30 meV is much larger than the exciton binding energy of about 6 meV, excitons are reaching down to scattered into the band-continuum state with a rate of reg (15Ofs)-' at room temperature. In almost all studies of the temperature dependence of the exciton spectrum, the spectrum width is given by adding up the FWHM of each broadening factor, that is, To 2hT,,. This is partly because the treatment is much simpler and partly because the origin of the inhomogeneous broadening is attributed to alloy or interface scattering. Since actual quantum wells have structural inhomogeneity on a much larger scale than that of the exciton radius, not all the broadening factors can have a scattering origin. Thus, convoluted integration of Eq. (1.189) is much more realistic.

+

5.

EMISSION OF PHOTONS IN QUANTUM WELLS MESOSCOPIC QUANTUM DISKS

SPONTANEOUS AND

In Section 111.2, we derived a formula for the lifetime of the electron-hole recombination through the spontaneous emission process. The resultant formula was independent of the dimension of semiconductor nano-structures except in the transition matrix, and gave nanosecond-order lifetime. As will be seen in this section, the exciton spontaneous emission property depends on the dimension and volume of the crystal and differs much from that of the electron-hole pair. This is primarily due to the selection rule to preserve the wave vector between the exciton center-of-mass motion and a photon. In bulk materials, this characteristic forms an exciton-polariton that propagates in the crystal. In quantum nano-structures, it is altered because the translational symmetry is lost in the quantum-confinement direction. Theories of exciton spontaneous emission lifetime in semiconductor nano-structures have been presented for quantum wells and quantum dots (Hanamura, 1988; Feldman et al., 1987; Andreani et al., 1991; Citrin, 1993; Sugawara, 1995). In quantum wells, the wave vector selection rule holds for the in-plane component, since the exciton motion is free in the twodimensional plane, and the lateral length of the quantum-well plane is much greater than the exciton resonant wavelength. As long as the emitted photon is not reabsorbed by the quantum well (this occurs in microcavities, as discussed in Section IV.6), the two-dimensional coherent nature causes rapid spontaneous emission on the order of picoseconds at low temperatures. In quantum dots, the spontaneous emission rate increases in proportion to the crystal volume. This occurs because, quantum dots have a size similar to the exciton Bohr radius and are much smaller than the resonant wavelength; thus, their electric dipole approximation of exp(iq.R) z 1 holds for the

56

MirsuRu SUGAWARA

center-of-mass motion, giving a transition matrix that is proportional to the volume covered by the exciton wave function extent (Eq. (1.172)). The expected crystal volume dependence is experimentally proved in CuCl microcrystals and is thought to be due to the coherent nature of excitons throughout entire microcrystals (Itoh, 1992). In this section, theoretical formulae for the spontaneous emission lifetime of excitons are derived in quantum wells and in mesoscopic semiconductor quantum disks. The mesoscopic disks approach macroscopic quantum wells as the disk diameter increases above the exciton resonant wavelength; they approach microscopic quantum dots as the radius decreases below the exciton Bohr radius. The lifetime formula of the disk indicates how the spontaneous emission lifetime varies from quantum wells to quantum dots. It is found that the lifetime increases two orders of magnitude as the disk radius decreases.

Excitons in Quantum Wells Let us consider a quantum a well with well width, L,, and an exciton resonance wavelength, Leg, that satisfies the following relation: Lz < 2u, < Leg << L

(1.192)

Here, uB is the exciton Bohr radius in bulk materials and L2 = D is the area of quantum-well crystals in the x-y plane. Under this condition, electron and hole energies are quantized in the z direction, and excitons move freely in the x-y plane. We also assume that the length of the crystal in the z direction, L, is much larger than A,,. The electromagnetic field is quantized in the crystal with the volume of V, = L3 and with the refractive index of n,. The basic rules to derive a formula of exciton spontaneous emission lifetime in quantum wells are as follows: (1) The selection rule of the in-plane wave vector is taken into account between the electromagnetic field, 411, and the exciton center-of-mass motion, KII, that is, KII = qll (Eq. (1.166)); (2) only excitons with the wave vector within a critical value, qc, can radiate spontaneously because of the energy conservation rule. The emission is in the direction of (q”, q,), with q, = ,,/(n,o,g/c)2 - q1Iz.This is illustrated in Fig. 1.5(b). The procedure to derive the formula is almost identical to the one used for the electron-hole pair in Section 111.2. The wave function for the exciton-photon system is written as

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES57

where I@,,,O) is the exciton state with no photon and lag,lq,b)is the ground steate with one photon. We assume the initial condition where one exciton with the resonant frequency of we, is in a quantum well and the field is in the vacuum state with a(0) = 1 and bq.b(0)= 0. The exciton drops into the ground state by spontaneously emitting a photon with an energy of wq and a wave vector of q. For the exciton-photon interaction Hamiltonian, Eqs. (1.159) and (1.165) with n = 0 are used. By substituting Eq. (1.193) into the Schrodinger equation of Eq. (1.63) and combining the equations of motion for a(t) and b,,(t), we get

where the coupling constant is (1.195) The rate, yd, includes all the dissipation processes besides the spontaneous emission, such as acoustic phonon scattering into nonradiative states in K space, ionization due to longitudinal optical phonons, and nonradiative recombination processes at crystal defects or impurities. As long as the exciton interacts with the continuous optical mode, Eq. (1.194) is reduced to Fermi's golden rule, as seen in Eq. (1.68). This is also the case for excitons in quantum wells, which interact with a definite optical mode in the in-plane direction and with continuous optical modes in the z direction. So, we again get Fermi's rule as b(t) =

'z,[

-

4t) + yd] 2

(1.196)

with 'si1

= 2'1

Igq,o12'(wq - w e g )

(1.197)

4."

By adding the two independent polarization components, the square of the coupling constant becomes

58

MITSURUSUGAWARA

Since exciton states are formed mainly from the states around the inverse of the relative motion radius, kll is replaced by’,1 in the polarization factor. Considering the optical-mode density in the z direction, as in Section 111.2, we obtain the spontaneous emission rate as

where ylI and q 2 are given in Eqs. (1.73) and (1.74). Comparing Eq. (1.72) for the electron-hole pair and Eq. (1.199) for the exciton, the most remarkable difference is the wave-vector selection rule. The spontaneous emission spectrum is given as

( 1.200)

where y = z,pl + y d . The spectrum is a Lorentzian function with an FWHM of y. Let us check numerically the spontaneous emission lifetime of the electron- heavy-hole exciton with K1l = 0 for the In,~,,Ga,,,,As/InP quantum well with a well width of L, = 10nm. Material parameters are m,*" = 0 . 0 5 ~ V~I $; =0.12m0; M:" =0.035m,,; M*" =0.17~,; M,*' =0.05mn; m t L = 0 . 5 ~and ~ ; n, = 3.5. The electron masses are calculated from the first-order k . p matrix and the hole masses from the Luttinger-Kohn parameters obtained by analyzing the diamagnetic shifts (Sugawara et al., 1993a,b). Equation (1.199) gives the lifetime of 23 ps, about two orders of magnitude shorter than that of the electron-hole pair transition. This fast emission is due to the selection rule to convey the oscillator strength of excitons into one optical mode with the in-plane wave vector fixed. However, the fast emission occurs only at low temperatures the exciton thermalization above the critical wave number of q, in the center-of-mass motion wave vector space. Since the energy of ti2qF/(2M*ll) is around 1 K, at which low temperature excitons are localized at some potential fluctuations, it seems almost impossible to observe a fast decay with the usual photoluminescence measurements. The effect of exciton localization on exciton's spontaneous emission lifetime can be modeled on the spontaneous emission in quantum disks, as discussed below.

1

OPTICAL PROPERTIES OF SEMICONDUCTOR

QUANTUM NANO-STRUCTURES 59

Excitons in Mesoscopic Quantum Disks

Let us discuss the spontaneous emission lifetime in quantum disks with a width of L,, which is less than 2u,, and a radius of R , (Fig. l.lO(a)). The disk consists of a direct band-gap semiconductor, and both an electron and a hole are confined in the disk by the potential barriers of surrounding regions. When R , is much greater than the exciton resonance wavelength in the crystal, Ae,/n, (that is, R , >> &Jn,), the disk is a quantum well with a well width of t,.When 2R, is approximately equal to L,, it is a quantum dot. The disk is mesoscopic when ,Ie, < R , < A,,/2nr. For example, A,, = 16.0 nm, A,, E 1.6 pm, and n, = 3.5 in a 10-nm In,,,,Ga,,,,As/InP quantum well. The origin of the coordinate axis is the center of the disk, and the z-axis is perpendicular to the layer. In the z direction, the disk has the same carrier confinement potential as that of quantum wells. The additional in-plane potential gives three-dimensional quantum confinement. The disk is situated inside the crystal with the size of L, which is much larger than the exciton resonant wavelength. The radiation field is quantized in the crystal with a volume of V , = L3 and a refractive index of n,.

,<

(a)

(b)

Disk radius. R," (nm) . . 200 100

50

20

Confinement potential,

10

6

w, (s-')

FIG. 1.10. Calculation of the exciton spontaneous emission lifetime (from Sugawara, 1995. Copyright 1995 by The American Physical Society). (a) Quantum disk with a width L,, which is less than 2u,, and a radius R,. The origin of the coordinate axis is the center of the disk, and the z-axis is perpendicular to the layer. (b) Calculated spontaneous emission lifetime (solid line) of the optically active ground-state excitons as a function of w, and R, by Eq. (1.208). The disk is In,~,,Ga, 4 7 A with ~ a width of Lz = IOnm, and the surrounding barrier is InP. As R, exceeds L,,/2n,, the lifetime gradually increases from a constant value. Under a strong confinement of R , < A,,, the lifetime increases by two orders of magnitude. The horizontal dashed line is the lifetime of free excitons with K" = 0 in quantum wells (23 ps), calculated from Eq. (1.199), and the dotted line is that calculated from the electric dipole approximation of Eq. (1.209).The lifetime calculated from Eq. (1.208) smoothly connects the two limiting cases.

60

MITSURUSUGAWARA

The disk is assumed to have the harmonic-oscillator-type in-plane parabolic potential described in Section IV.2. Let us expand the localized center-of-mass motion wave function of Eq. (1.144) into a Fourier series to apply the selection rule with the radiation field as

(The subscripts for the quantum numbers are added for clarity.) The expansion is done in the whole crystal. From the periodic boundary condition, KII = 27c(n,/L,n,/L), where ni is an integer (i = x and y). Note that Eq. (1.201) is the superimposition of free-exciton wave functions with various wave vectors. For the optically active excitons with 1 = 0 (see Eq. (1.204) below), the Fourier coefficient in Eq. (1.201) is given as

p exp( - fl2K1I2/4)Lk(p2;Il2) ~

k!

(1.202)

where L, is the Laguerre polynomial. Equation (1.202) shows that, as fl decreases, excitons become composed of the states with larger wave vectors. For the ground state of k = 0, Eq. (1.202) gives

C,,(K~~)=

&

p exp( - p2K1I2/4)

(1.203)

Using the relationship of (1.204) the disk radius is defined as

R,

=

*fl=

2,/-

since 2np2 is the area covered by the center-of-mass motion. The square of the coupling constant is

(1.205)

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 61

Note that in & I , ~ , I Eq. (1.198), for free excitons is replaced by lCk,(qll)lz. Using the same procedure as before, the spontaneous emission rate on the basis of Fermi's golden rule is given as

Using Eq. (1.202) for optically active excitons with 1 becomes

= 0,

Eq. (1.207)

For the ground-state excitons (k = 1 = 0), Lk(P2q2/2)/k!= 1. Equation (1.208) overlaps with the formula for quantum dots as fi decreases; that of quantum wells, as p increases. When &,<< 1, that is, j << Aex/nr, exp( -p2q2/2) 2 1 - pzq2/2 z 1, and L,(P2q2/2)/k ! 1. Then, Eq. (1.208) reduces to

The spontaneous emission rate is proportional to the area covered by the center-of-mass motion. We can derive Eq. (1.209) using the three-dimensional mode density of the electromagnetic field quantized in the crystal with the refractive index of n,, and using the electric dipole approximation of exp(iq-R) = 1, which is valid for quantum dots much smaller than the resonant wavelength. When the variable separation is prohibited, we have only to replace the wave function as (1.210) where $ex(r, R) is the three-dimensionally confined exciton wave function. For large p, we can substitute tl,(O), qz(0), and Lk(0)in the integral, so that

62

MlTSURU SUGAWARA

Eq. (1.208) reduces to

represents the spontaneous emission lifetime of free excitons where z(,O) with K" = 0 in quantum wells (Eq. 1.199). Thus, as ,f? increases, Eq. (1.208) approaches that for quantum wells. As a result, Eq. (1.208) for the exciton spontaneous emission lifetime continuously connects the microscopic and macroscopic limits via mesoscopic regions. In Fig. l.lO(b), the solid line is the calculated spontaneous emission lifetime of the optically active ground-state excitons as a function of w, and R , using Eq. (1.208). The disk is In,,,,Ga,,,,As; the width is L, = lOnm, and the surrounding barrier is InP. Since the topmost valence band is the heavy-hole band in this sytem, an electron-heavy-hole exciton is considered. As R , exceeds IZeg/2nr,the lifetime gradually increases from a constant value. This is because the center-of-mass motion wave function in K space spreads outside q,, and the components that never contribute to spontaneous emission increase. Under a strong confinement of R , < A,,, the lifetime increases by two orders of magnitude, The horizontal dashed line is the lifetime of free excitons with K = 0 in quantum wells (1 1.5 ps) calculated from Eq. (1.199), and the dotted line is that calculated from the electric dipole approximation of Eq. (1.209).The lifetime calculated from Eq. (1.208) smoothly connects the two limiting cases. Temperature Dependence of the Exciton Lifetime

The spontaneous emission lifetime varies with temperature due to the thermal distribution of excitons in K space. The way the lifetime depends on temperature varies from quantum wells to quantum dots. In quantum wells, the distribution of excitons outside q, increases the average exciton lifetime. This principle for the lifetime increase is similar to the case of an exciton in a quantum disk. Assuming that the thermal distribution in Kll space obeys the Boltzmann distribution, and averaging the lifetime over K" space, the temperature dependence of the exciton lifetime is given as ts;'(T) z z,pl(0)[l - exp(- T,/T)] 2 Tsi'(o)(Tc/T)(T >>

T)

(1.212)

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 63

where T, = h2q2/2M*llk,. Since T, = 0.6 K in In,~,,Ga,,,,As, the lifetime increases almost linearly with temperature. In quantum disks, using the discrete energy levels of Eq. (1.140), the temperature dependence of the lifetime for the ground-state excitons is written as

where AEf, = E i - Et0 and AEL", = E*,, - E\o. The denominator represents all the carrier distribution, and the numerator represents the carrier distribution only to the optically active states. The lifetime increases as the temperature increases due to the distribution of excitons to the inactive states. As the quantum-disk radius decreases, the lifetime becomes insensitive t o temperature due to the increase in the interlevel separation. In quantum dots with an energy separation much larger than k,T = 25 meV, the lifetime is kept almost constant up to room temperature. E.uperirnents on the Relationship Between Exciton Localization and Spontaneous Emission

Theoretical consideration of spontaneous emission in quantum disks indicates that, if excitons are localized in quantum wells at low temperatures, their spontaneous emission lifetime drastically increases. Experimental evidence is provided here by comparing the spontaneous emission properties between free and localized states. Figure 1.1l(a) shows the photoluminescence spectra of Ino,,,Ga,,,,As/ InP quantum wells, which can be attributed to the spontaneous emission of the ground-state electron-heavy-hole excitons. Note that the peak wavelength and intensity of the exciton spectra depend on temperature in an unusual manner. While the peak wavelengths are almost the same at 2 K and 4.2K, the spectrum at 20K is found at a shorter wavelength. The intensity at 4.2K is slightly higher than that at 2 K . As temperature increases above 20 K, the spectrum gradually shifts toward longer wavelengths and its intensity decreases. The photoluminescence spectrum at 2 K appears in the lower energy of the exciton absorption resonance by a Stokes shift of 4.5 meV. Figure I.ll(b) shows the diamagnetic shifts of exciton spectra up to 12 Tesla for photoluminescence at 2,4.2,40, and 100 K, and optical absorption

64

MITSURUSUGAWARA 10

.2. c u)

2

c .-

8

5

05

0c

.g0 0

I

E o

1 42

(4

146

150

Wavelength (pn)

Free excitm

0;

(b)

Magnetic field (Ta)

(c)

A

40

M,

'

eb

Temperature (K)

’A

FIG. 1.11. Experiments on exciton spontaneous emission (from Sugawara, 1995. Copyright 1995 by The American Physical Society). (a) Photoluminescence spectra of 10-nm In,,,,Gao,,,As/InP quantum wells resulting from the spontaneous emission of the ground-state electron-heavy-hole excitons. (b) Diamagnetic shifts of exciton spectra up to 12 Tesla for photoluminescence at 2,4.2,40, and 100 K and optical absorption at 4.2 K. The magnetic field is applied perpendicular to the quantum-well layers. The error bars represent the shifts of self-assembled In,~,,Ga,,,As quantum dots in Fig. l.l(b). Values of w, are indicated in Fig. l.lO(b). (c) Peak intensity of photoluminescence as a function of temperature. The intensity of localized excitons (below 20 K) is almost constant; the intensity of free excitons decreases as temperature increases. The exciton localization significantly enhances the spontaneous emission lifetime.

at 4.2 K. The magnetic field is applied perpendicularly to the quantum-well layers. While the shifts of photoluminescence at 40 K and 100 K almost agree with those of the optical absorption, they are clearly smaller at 2 and 4.2 K. Those at 20 K are in between (not shown). The 2- and 4.2-K shifts can be attributed to the presence of additional in-plane confinement potential. The error bars represent the shifts of the self-assembled In,,,Ga,,,As quantum dots of Fig. l.l(b). Very small diamagnetic shifts

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 65

clearly demonstrate their strong three-dimensional quantum confinement. The lateral confinement of excitons at low temperatures in the quantum well is weaker than that of excitons in the quantum dots. The solid lines show the calculation of the diamagnetic shifts by

V,II

-

]

e2 ( e 2 B 2 I rn:lk.o~) rli2 Yr = E,Y, (1.214) 4 7 - c ~ 8m,*11 ~ ~ ~ ~2 ~ ~

+-

with m,*" = 0.035m0 and E = 13.9~,,using o,as a parameter. The harmonicoscillator potential is added to Eq. (1.150). Comparing the calculation with the measured shifts, we find that excitons at 2 and 4.2K are laterally confined with the potential of w, = 1 x 1013s-'. We obtain = 11.6 nm for w, = 1 x 1013s-' using M*" = 0.17m0. The radius of the relative motion wave function at a zero field decreases from Aex = 16.0 nm to 11.3 nm due to lateral Confinement. At 40K and 100K, the confinement potential is below the detection limit of 1 x 10" s-' and the excitons are almost free. The diamagnetic shifts in Fig. 1.1 l(b) indicate that localized excitons below 20 K are subject to intermediate lateral confinement between free excitons in quantum wells and three-dimensionally confined excitons in quantum dots. Figure 1.1l(c) plots the peak intensity of photoluminescence as a function of temperature. Note that while the intensity of the localized excitons (below 20K) is almost constant, the intensity of the free excitons decreases as the temperature increases. We analyze exciton photoluminescence intensity using

(1.215) where G is the generation rate of excitons by laser excitation, T~ is the radiative spontaneous emission lifetime of free or bound excitons, and znr is the nonradiative recombination lifetime. The solid line in Fig. l.ll(c) is calculated from Eqs. (1.212) and (1.215) to fit the data for free excitons / T0.1 ~ ~ and G = 12 in arbitrary units). It is assumed above 20 K ( T ~ ~ ( O ) = that the nonradiative recombination lifetime is independent of temperature. The calculation indicates that, if the excitons were free even below 20K, their intensity would continue to increase and become much stronger than that of the localized excitons. The localized exciton emission intensity (the horizontal dashed line) indicates that the localized exciton lifetime is zB = 32tS,(0); that is, exciton localization significantly enhances spontaneous emission lifetime. The solid circle in Fig. 1.10(b) represents the

66

MITSURUSUGAWARA

localized exciton lifetime in In,,,,Ga,,,,As/InP quantum wells, plotted = 1 1 . 5 ~The ~ . estimusing o,= 1 x 1013s-", z B = 32rS,(0),and z,(O) ated lifetime agrees quite well with the calculation, supporting the theoretical prediction.

OF PHOTONS IN QUANTUM DISKS 6. SPONTANEOUS EMISSION PLACED IN A PLANAR MICROCAVITY

The exciton-photon interaction in semiconductor microcavities has attracted much attention since the discovery of coupled-mode splitting between cavity photons and resonant excitons in quantum wells (Weisbuch et al., 1992). The coupled mode is called a cavity polariton, from the analogy of exciton polaritons in bulk semiconductors (Houdre et al., 1994), or dressed excitons, from the analogy of dressed atoms in microwave cavities (Cao et al., 1995). Many experiments in the spectral region, such as the reflection, transmission, and photoluminescence spectra, not only revealed how energy splitting depends on exciton-cavity-photon mode detuning, oscillator strength, and quantum-well position in the cavity, but also clarified exciton-polariton dispersion relations (Houdre et al., 1994; Zhang et al., 1994). Also found was the oscillatory behavior of spontaneous emission in the temporal region, which is analogous to Rabi oscillations in atomic physics and is evidence for the coherent exciton-photon energy exchange (Cao et al., 1995; Norris et al., 1994; Jacobson et al., 1995). These phenomena contribute greatly to semiconductor physics in that we are now able to control exciton-polariton behavior by artificially designing microcavity structures. This might lead to new types of semiconductor devices, such as THz electromagnetic field generators (Kadoya et al., 1995). As we saw in Section IV.5, exciton spontaneous emission occurs by preserving the wave vector between the exciton center-of-mass motion and the emitted photon. In quantum wells, the wave vector selection rule for the quantum-well plane component gives rise to fast decay with a spontaneous emission lifetime of about lops. The decay is exponential due to the exciton's coupling to the z-direction continuous optical mode. If the zdirection continuous photon mode is changed into discrete modes using a planar cavity, one-to-one correspondence is realized between the exciton and photon modes for all directions, leading to strong coupling where the coherent exciton-photon energy exchange occurs. Here, the mesoscopic semiconductor disk in Section IV.5 is placed in a planar cavity to observe the cavity effect on the exciton spontaneous emission. Excitons resonant with a cavity-photon mode show cavity-enhanced spontaneous emission. The enhancement is more remarkable as the disk

1

OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 67

radius increases and the exciton's in-plane coupling to the photon mode becomes stronger. When the disk radius is much larger than the exciton resonant wavelength and the cavity photon decay rate is smaller than the exciton-photon coupling constant, the exciton-photon system transits from the weak coupling to the strong-coupling regime, leading to oscillating spontaneous emission decay. The temporal evolution of the emission spectrum is evaluated to show coupled-mode spectrum splitting. Figure 1.12(a) shows the coordinate system, where the x- and y-axes lie in the mirror plane and the z-axis is normal to the mirrors. The mirrors are assumed to be spaced a distance, L , apart, and have an area, D = L2, where the length of mirrors in the x and y directions is infinite compared to the exciton resonant wavelength, ;leg. The planar cavity is assumed to be ideal in the sense that it has a reflectivity, R,, that is constant not only as a function of wavelength but also as a function of the incident angle. The radiation field is quantized in the cavity with a volume of V , = L L 2 and a refractive index of n,. The quantum disk is placed in the middle of the cavity with its center at z = 0. Figure 12(b) represents the wave number space of the cavity-disk system. Due to the resonant condition, only excitons with an in-plane wave number less than q, = n,o,,/c can radiate spontaneously. When the wave number of the cavity mode in the z direction is fixed at q, and the resonant condition of weg= c(ql12 + qZ)'i2/n, is satisfied, the excitons emit photons in the direction of (K1l,qz). For real Bragg mirrors with alternate high and low refractive indices, the photon emitted in the direction outside a critical angle connects with the electromagnetic-field background in free space. This means that excitons with the in-plane wave vector below a critical value, qs, are connected into the cavity mode and excitons with qs < K d y, are connected into the open-side mode. It is assumed that the photons repeatedly reflected between the mirrors interact with the disk every time they pass the middle of the cavity. This assumption is particularly important when the cavity photon proceeds in the declined direction and the disk radius is smaller than the resonant wavelength. Here, to satisfy this condition, it is assumed that disks are packed in the z = 0 plane to the extent that the distance between neighboring disks is significantly smaller than the exciton resonant wavelength. However, the overlap of exciton wave functions between neighboring wells is still assumed to be negligible. The photons emitted in the declined direction due to the nonzero in-plane wave vector component depart from the initial disk and interact with another one. This is the exciton transfer between the disks in the in-plane direction. The inclusion of the cavity effect in the exciton decay problem is not straightforward, although the basic strategy to expand the exciton wave

68

MITSURUSUGAWARA

(a

Disk radius, Ro (nm)

FIG. 1.12. Calculation of exciton spontaneous emission in the microcavity (from Sugawara, 1997h). (a) Quantum disk placed in a planar cavity. The x- and y- axes lie in the mirror plane, and the z-axis is normal to the mirrors. The quantum disk is placed in the middle of the cavity with its center at z = 0. The mirrors are assumed to be spaced a distance, L', apart and have an area of D = Lz, where the length of the mirrors in the x and y directions is infinite compared to the exciton resonant wavelength, I,,,. (b) Spontaneous emission into the cavity mode with the wave vector q = (qll, qz),where q" is equal to the exciton center-of-mass motion wave vector KI1. When 4. < K" i4,. the emission is into the open-side mode. (c) Spontaneous emission lifetime of excitons in the 10-nm In,,,,Ga,,,,As/InP system as a function of the disk radius, R, =b , /3. The longitudinal dotted lines indicate half of the exciton resonant wavelength, ieX/(2nr)P 200 nm, and the in-plane exciton radius for quantum wells, lex = 16.0 nm. The dashed line is for free space without a cavity. The solid lines are for the cavity with a variety of photon lifetimes, K-’.

1

OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 69

function in K" space need not be changed. First, the z-direction opticalmode density should be changed from the continuous one in a free space to a definite one in a cavity. Second, the formula for the coupling constant should be modified to include the standing-wave electric field in the cavity. Finally, instead of the simple Fermi's rule, a general expression that is valid regardless of coupling strength, should be used, since the coupling strength varies continuously from the strong-coupling regime to the weak-coupling regime depending on the disk radius. All of these processes bring considerable changes in the resultant formulae, as shown below. The exciton decay in a planar microcavity is typically characterized by the following coupling constants: the dipole coupling constant between the exciton and the cavity-mode photon, y; the decay rate of the cavity-mode photon via mirror losses, K ; the spontaneous emission rate into the openside modes due to the electromagnetic field background, yo; and the exciton dissipation rate besides spontaneous emission, yd. The photon decay rate is given as K =

C

---lnR, n, L’

(1.216)

As shown in Fig. I.l2(b), yo depends on the critical in-plane wave vector, qs, above which excitons are connected to the open-side mode. The vector potential in a planar cavity is modified from Eq. (1.59) by internal reflection and interference. For the S-polarized mode (its electric vector is normal to the plane defined by q and the z-axis), we obtain

For the P-polarized mode (its electric vector is in the plane defined by q and the z-axis), we obtain

x

[&4+

a,(t)

+ e-'q"i?:(t)]

(1.2 18)

The z-direction wave vector is given as qz = mn/L'

( m = 0,1,2,. . .)

(1.219)

and the frequency of the cavity-photon mode is

(1.220)

70

MITSURUSUGAWARA

The wave function for the exciton-photon system is the same as in Eq. (1.193). Again, the initial condition is that one exciton is in a quantum disk and the field is in the vacuum state with a(0) = 1 and b,,(O) = 0. The exciton drops into the ground state by spontaneously emitting a photon with an energy of ho, and a momentum of hq. Using the rotating wave approximation, the equations of motion for the probability amplitude of a(t) and b,,(t) are found as

and (1.222) Taking into account both P and S polarization, the coupling constant is given as

where

The upper term in parentheses corresponds to the free excitons in quantum wells, and the lower term corresponds to the localized excitons in disks. From Eqs. (1.221) and (1.222), as before, we obtain

Depending on the way the exciton interacts with the electromagnetic field, Eq. (1.225) gives the following types of solutions: Decay into the open-side mode in the cavity: The components with the wave vector in the range of q, < q d q, contribute to this decay with a rate of yo.

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 71

Weak coupling with the cavity mode: For K >> g, the emitted photon escapes from the cavity and is barely reabsorbed by the quantum disks. In this weak coupling regime, a ( t ) can be removed from the integral. Thus, the remaining integral gives, for t >> C1, (1.226) where

and ( 1.228)

is the normalized Lorentzian function representing the cavity-mode broadening with the FWHM of K. Equation (1.226) gives the exponential decay as in the free space. The validity of the weak-coupling formula is ensured if the calculated result of zspl is sufficiently smaller than K . This corresponds to the situation in which the cavity-mode broadening of Eq. (1.228) is greater than the homogeneous broadening of exciton emission given by Eq. (1.200) and the optical mode is considered to be still continuous. If K is nearly equal to or less than z5G1, we should resort to the general solution below. Strong coupling with the cavity mode (a general case in the cavity): When the coupling between the exciton and the cavity is so strong that a photon emitted into the cavity is likely to be absorbed before it escapes, the approach in the weak-coupling regime ceases to be justified (Goldstein and Meystre, 1995). In this case, since a(t') changes more rapidly than the exponential dissipation factor, we should numerically evaluate Eq. (1.225). By transforming the variable of the integral, we obtain a(t

-

s)f(s)eCKSI2 ds -

(t+ $)

a(t)

(1.229)

72

MITSURUSUGAWARA

where

Analytical expressions for the strong-coupling limit can be obtained only for free excitons in quantum wells that interact with a definite cavity mode. By omitting the q summation in Eq. (1.221), the second derivative equation on a(c) can be derived. The solution to the equation is written as a(t) = clen"

+ c2en2'

(1.231)

where c(1,2=

Q2 =

(y,

- - -+-++ IC( j ,

2 2

2

1

2

) +' [(h---2

2

ti

2

i(jqy -

lgq,aI2

~cP]'"

(1.232) (1.233)

o=P.S

6, = oq- ueg, q = (Kll, qz), and cl.z are numerical coefficients determined by the initial conditions of a(0) = 1 and h(0) = -yd/2. The factor of two in Eq. (1.233) represents the two z directions, that is, the front and back sides of the cavity. For resonant excitons with 6, = 0, Eq. (1.231) becomes

cos(@t/2)+ -sin(Qt/2)] 20

(1.234)

with (1.235) and with the strong-coupling condition of 4 0 - ( y d - ti)2/4 > 0

(1.236)

The frequency of Q is known as the Rabi frequency in atomic physics.

1

OPTICAL PROPERTIES OF SEMICONDUCTOR

QUANTUM NANO-STRUCTURES73

From Eqs. (1.221) and (1.222), the coefficient for each optical mode is given as

The emitted photon power at a time, f, and at a frequency between w and w + dw is given as

_-

1

(1.238)

Equation (1.238) gives the temporal evolution of the emission spectra. Numerical Calculations Let us use the half-wavelength cavity with the cavity space as L' = Aeg
<

q;

2nmn, =

~

&g

i

(rn

= 0,l)

(1.239)

Only the m = 1 mode is taken into account since the rn = 0 mode is coupled to the exciton with q = q, (>q,) and should be treated as the open-side decay in a real planar microcavity. For simplicity, the dissipation rate into the open-side mode is not taken into account here. The spontaneous emission of the optically active ground-state excitons is calculated for the In,,,,Ga,,,,As/InP disk with a thickness of L, = 10nm. Free excitons in quantum wells have a resonant wavelength of Leg = 1.45 pm, a resonant frequency of weg= 1.30 x l 0 l 5 s-' at 4.2K, and an in-plane radius of 16 nm. The exciton resonant wavelength in the cavity is approximately given as Aeg/nr2 400 nm. The dissipation rate of yd is neglected so that the cavity effect can be clearly seen.

14

MITSURUSUGAWARA

For K1I= 0 excitons in the exact half-wavelength cavity of [ = 1, Eq. (1.227) becomes

( 1.240)

The rate is 4oeg/lc71times larger than that of K = 0 excitons in a free space given by Eq. (1.199). This effect is well known as cauity-enhanced spontaneous emission. The solid line in Fig. 1.12(c) represents the spontaneous emission lifetime of Eq. (1.227) into the m = 1 mode as a function of the disk radius for the exact half-wavelength cavity (i= 1) with K - ~= 240fs, 51 fs, 23 fs, and 11 fs ( R P= 0.99, 0.95, 0.9, and 0.8, respectively), and the dashed line is the lifetime in free space. In quantum wells-disks with a radius much larger than the exciton resonant wavelength- the emission rate is enhanced by about two or three orders of magnitude from that in a free space as the photon lifetime increases. In the cavity with K - = 240 fs, the exciton lifetime is reduced below the photon lifetime, and the transition from the weak-coupling regime to the strong-coupling regime occurs. Needless to say, Eq. (1.240) can not be applied in this strong-coupling regime. It is clear from Eq. (1.227) that the emission of excitons with KI1 > 0 is suppressed since no optical mode is available. This condition is known as inhibited spontaneous emission. The lifetime increases from a constant value with the decrease in R, because localized excitons in quantum disks spread KII-space wave functions and connect only weakly to the cavity mode. Note that, while the emission rate of free excitons in quantum wells is enhanced by two or three orders of magnitude due to the cavity effect, the rate in the dots is hardly changed by the planar cavity. Only about a 30% increase in the emission rate is expected even under a very small photon decay rate. This weak coupling to the cavity mode comes from the extended exciton wave function in K" space. To realize the strong coupling between excitons in quantum dots and the photon mode, we need to control the cavity photon mode three-dimensionally. Figure 1.13(a) represents la(t)12 for K" = 0 excitons in the exact halfwavelength cavity as a function of t for K-’ = 240 fs, 80 fs, 51 fs, 23 fs, and l l f s ( R P = 0.99, 0.97, 0.95, 0.9, and 0.8, respectively), with yd = 0. The minimum point of the oscillation should approach zero, and its finite value merely comes from the finite discrete time interval used in the calculation. While the almost exponential decay occurs at K - = ~ 23 fs and 11 fs, the oscillatory behavior appears at K-’ = 240 fs and 80 fs. In the intermediate ~ 51 fs, the curve is not completely exponential at the initial case of K - = decay of 0 to 200fs. The oscillation approaches la(t)12= cos2(Rt) as the

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 75

(a)

(c)

Time (ps)

Time (ps)

FIG. 1.13. Calculation of exciton spontaneous emission in the microcavity (from Sugawara, 1997). ( a ) Probability of ln(t)12 for K = 0 excitons in the 10-nm In, ,,Ga, ,,As/InP quantum = I ) are wells as a function of r. The photon lifetimes in the half-wavelength cavity (i ti-’ = 240fs, 8Ofs, 51 fs, 23fs, and 11 fs. (b) Probability of la(f)I2 for K = 0 excitons in the 10-nm In,,,,Ga,,,,As/InP quantum wells when I < - ' = 240fs at a variety of detunings i. (c) Probability of la(t)12 for the ground-state excitons in the In, ,,Ga,,,,As/InP quantum disks in the half-wavelength cavity (( = 1) when ti- = 2.4 p.

'

photon lifetime increases. For K = 0 resonant excitons in the halfwavelength cavity, Eq. (1.233) gives

(1.241)

The cavity-exciton detuning dependence of exciton emission decay is shown in Fig. l.l3(b), which describes the temporal evolution of lu(t)12 for KII = 0 excitons when IC-’ = 240 fs at a variety of detunings, i.Note that

76

MITSURUSUGAWARA

the m = 1 mode frequency is weg/(.As increases, the contrast of oscillatory decay becomes unclear and the oscillation frequency increases. Even at 0.5% detuning, the oscillation disappears after a few repetitions. Figure 1.13(c) represents la(t)12 in the exact half-wavelength cavity as a function o f t for the disk radius of R , = 7400, 2300, 1200, 400, and 200nm when K - = 2.4 ps. As R , decreases, the exciton wave function spreads in K apace and the exciton-cavity-mode coupling is weakened, resulting in a decrease in the oscillation frequency and the oscillation amplitude. At around 400 to 1200 nm, the oscillation gradually disappears, as expected, from the weak-coupling calculation in Fig. 1.12(c). Considering that the exciton resonant wavelength is Leg/nr z 400 nm, the radius should be larger than the resonant wavelength to exhibit the cavity polariton, that is, Ro > L e g l n r . Figure 1.14 shows the temporal evolution of the emission spectra for (a) R, = 200 nm and (b) R , = 7400 nm, where IF(w - oeg, t)I2 in Eq. (1.238) is plotted as a function of w - we, for 4' = 1 and K - ' = 2 . 4 ~ For ~ . R, = 200nm, where the almost exponential decay is observed in Fig. l.l3(c), we obtain emission spectra at w = w,, - 0.3 [THz] with a fullwidth at half maximum proportional to the inverse of t . Considering that the exciton resonance frequency is weg= 1.30 x 1015s-', the detuning of the emission corresponds to about 2.3 x 10-40,g and is almost impossible to detect in experiments. For R , = 7400 nm, we observe double peaks around o = weg separated by 2!2 = 8.4THz, where SZ is the oscillation frequency calculated from Eq. (1.241). Note that almost one oscillation period (0.75 ps) is necessary to observe the double peaks, showing that the coupled state-

101

(a)

'

.

'

'

!

, ' ' '

I

,

'

'

.

I

'

Detuning, o--oeg (THz)

'

'

'

1

(b)

Detunlng, o--ww(THz)

FIG. 1.14. (F(w- weg,t)I2 in Eq. (1.238) as a function of Q - w , for ~ (a) R , = 200nm and (b) R , = 7400nm, representing the temporal evolution of the emission spectra calculation on exciton spontaneous emission in the microcavity (from Sugawara, 1997b).

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 77

the cavity-polariton -is formed through one period of energy transfer between an exciton and a photon. The condition for excitons to show reversible spontaneous emission is summarized as follows. First, excitons should interact with a definite optical mode. This condition is satisfied for free excitons with a definite wave vector, K", in quantum wells, since they interact with only one photon mode with the wave vector (K1I,qz).This is due to the in-plane selection rule and the cavity-induced z-direction discrete optical mode. As the disk radius decreases below the exciton resonant wavelength, excitons begin to interact with continuous photon modes due to the spread of the wave function in K space, resulting in the destructive interaction between different modes leading to an exponential decay. Second, the photon decay rate should be smaller than the exciton emission rate; otherwise, the optical mode of Eq. (1.228) is broader than the homogeneous broadening of exciton resonance, which is equivalent to the situation in which excitons couple with continuous optical modes. Third, the exciton resonant frequency should closely match the cavity-mode frequency within 2R, since the detuning causes destructive interference in the time domain. To summarize, the conditions are (1) R , > Leg/nr; (2) zs;' > rc; and (3) Iw, - wegl/oeg < 2Q/w,,, where w, is the cavity-matching frequency. Both the homogeneous broadening due to yd and the inhomogneous broadening due to structural imperfections smear the oscillation, since they enhance the detuning from the cavity-photon mode.

7. THECOULOMB EFFECTON OPTICALGAINSPECTRA That excitons take part in lasing has long been discussed in wide band gap materials like 11-VI semiconductors with large binding energies (Koch et al., 1978, Haug and Koch, 1977, Klingshirn and Haug, 1981). This topic is once again of broad interest, given that green to blue semiconductor quantum-well lasers have been produced primarily owing to the success of high-density p-type doping. Room-temperature operation of ZnSe-based (Haase, 1991) and GaN-based (Nakamura et al., 1996), lasers has had a great impact on semiconductor optical technology, since short-wavelength lasers are the key to next-generation high-density optical data storage systems. Studies of the lasing mechanism in wide-gap quantum-well lasers have become critical in the creation of practical lasers and in their improved performance. Exciton-related lasing in 11-VI wide-gap semiconductor quantum wells was first suggested by experiments showing that lasing in ZnCdSe/ZnSe multiple quantum wells occurred around the tail of the free-exciton optical

78

MITSURUSUGAWARA

absorption resonance (Ding et al., 1992, 1993). Ding and his colleagues proposed a mechanism by which excitons created by photo excitation diffuse into the low-energy tail of inhomogeneous broadening and cause population inversion, leading to lasing. Experimental data have since suggested various kinds of exciton-related lasing mechanisms in quantum wells (for example, Kuroda et al., 1992; Jen et al., 1993; Kawakami et al., 1994; Masumoto et al., 1993; Kreller et al., 1995; Kozlov et al., 1996). We should discount the simple explanation that the large exciton-binding energy of wide-gap materials stabilizes exciton states, resulting in exciton lasing. This is because excitons as bound states between an electron and a hole cannot exist under the population inversion, as the semiconductor Bloch equation predicts (Haug and Koch, 1994; Uenoyama, 1995). Thus, in order for bound-state excitons t o generate a gain, the exciton emission energy should be shifted from the resonant exciton absorption energy through some mechanism. Possible emission processes are: (1) radiative recombination assisted by exciton-exciton, exciton-optical phonon, and exciton-electron (hole) scattering, (2) exciton localization; and (3) bi-exciton-exciton transition. All of these processes need the participation of a third partner besides the photon and excitons: scattering particles; local potential minima; bi-excitons. In this sense, exciton lasing occurs in very special circumstances. See Galbraith, 1995 on the calculation of the scattering-related mechanism. He showed that, in the ZnSe bulk active layer, the exciton-electron scattering process can cause a room-temperature optical gain of up to 100cm-’ at a lower carrier density than the . calculation suggests that Mott density of about 5 x 1 0 ” ~ m - ~This exciton-related lasing is possible if the cavity loss is well restricted with a high-quality long cavity and high-reflectivity mirrors. Lasing due to biexciton-exciton transition is discussed in Section V.2. Though exciton lasing will occur only in special cases, the Coulomb interaction itself considerably affects optical gain. Here, the semiconductor Bloch equation is summarized to see how the optical gain spectrum in Section 111.1 is changed by the Coulomb interaction. The advantage of Coulomb enhancement of gain for wide-gap laser performance is discussed.

Outline of the Semiconductor Bloch Equation In Section 111.1, the optical susceptibility of semiconductors was derived using a density-matrix equation where no electron-hole Coulomb interaction was taken into account. As the carrier density increases, the population factor of .f;,.k- fc,k becomes negative, generating the optical gain (Eq. 1.42)). In Section IV.4, the exciton optical absorption spectrum was derived using

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 19

a similar density-matrix equation with the exciton state as a base function. However, this treatment holds only for low or intermediate excitation where the interaction among excitons is so weak that the exciton base function can be used safely. Even so, in the high-excitation regime where the system transits into the electron-hole plasma state, the treatment breaks down. So, in order to describe the Coulomb effect on optical gain, a more general treatment is necessary where the base function of the system is not assumed a priori. The semiconductor Bloch equation describes the continuous variation of optical susceptibility from the low-excitation exciton regime to the highexcitation electron-hole plasma regime (Haug and Koch, 1994). The strategy to obtain a gain formula is to solve the Heisenberg equation of (1.242) where He, is the unperturbed electron and hole Hamiltonian of Eq. (1.115) and H I is the Hamiltonian resulting from the interaction with the photons of Eq. (1.152). The polarization of the system is given as

4

P(t) = - Qd d3r($+(r, t)ee; r$(r, t))

(1.243) where ( 1.244)

is called a pair function and Qd is the inverse of the crystal volume as defined before. Here, the two-band model and the k-selection rule are adopted, as in Section 111.1. The problem is to determine the pair function of Eq. (1.244) by solving the Heisenberg equation of Eq. (1.242). The four-operator terms coming from the Coulomb interaction term in Eq. (1.242) are approximately factorized into the products of the two operators: one two-operator term describing the carrier density and the other describing the interband pair function. The product of two operators with different wave vectors vanishes by taking the time average since they have different rotation frequencies. The semiconductor Bloch equation thus derived represents the kinetic motion of the pair function under the quasi-equilibrium condition. Its

80

MITSURUSUGAWARA

Fourier-transformed equation is

(1.245) where

represents the electron (i = ej and hole (i = h) frequencies red-shifted from the band-edge frequency of w,,k due to the screened exchange, C,,, and Coulomb-hole self-energies, ZcH; 6 represent the relaxation rate of the pair function; and Y k - k l is the Fourier-transformed screened Coulomb potential with a positive value. The Fourier transformation of the polarization is given as

and can be solved by Equation (1.245) is the simultaneous equation of Qcrrk, numerically inverting a matrix. It is clear that the Coulomb interaction in the nondiagonal terms mixes different wave vector states, giving rise to variation in the polarization. Figure 1.15(a) shows the calculated optical gain spectra for 6-nm GaNAle. 14Ga0,86Nquantum wells for various carrier densities (Chow et al., 1996). The exciton optical absorption resonance appears at a low excitation of 1 x 10’2cm-2, and the optical gain appears in the tail of the exciton resonance at high excitation. The gain peak is red-shifted with respect to the first absorption resonance. The shift is the net result of energy shifts, both due to band-filling, giving a blue shift that depends on the band structure and, due to the band-gap renormalization, giving a large red shift due to the &,. Figure 1.15(b) shows the gain spectra at the carrier term of C,, density of 7 x 1012cm-’ with (solid curve) and without (dashed curvej Coulomb effects, showing that the gain is enhanced as a result of the Coulomb interaction. This is called the Coulomb, or excitonic, enhancement of gain. In order to know what the Bloch equation means, let us rewrite Eq. (1.245) by setting the k’ summation of the product of the Coulomb potential and the pair function as

+

(1.248)

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 81 I

'E -1

I

I

I

E 0

0 U

m 0 -12

0 r

F

v

w

.-c -3

d

-5 -0.1 (a)

0.0 fi%-Ego

0.1

(ev)

0.2

-28 -0.1 (b)

0.1

h0-Ea

0.3 (ev)

FIG. 1.15. (a) Computed TE gain spectra at T = 300 K for 6-nm GaN-AI, ,,Ga,,s6N at various carrier densities. (Reprinted with permission from Chow et al., 1996. Copyright 1996, American Institute of Physics.) (b) Gain spectra at the carrier density of 7 x l O " c r ~ - ~ computed with (solid curve) and without (dashed curve) Coulomb enhancement. E,, is the unexcited bulk material band-gap energy. (Reprinted with permission from Chow et al., 1996. Copyright 1996, American Institute of Physics.)

The reason for Eq. (1.248) is that, the mixing of different wave-vector states under the Coulomb interaction, causes QL.,ck to spread in k space and there is a finite range around k, where Qu,ck can be regarded as almost constant. The term V, represents the summation of t $ - k / in the area and has a positive real value. The term V, is the rest of the summation of Yk-k,Qc,ck(u)on k'. When the Coulomb interaction is zero, that is, Yk-k'l = 0, only a resonant term of Qc.,ck responds to the electromagnetic field. Due to the mixing of different k states through the Coulomb interaction, the nonresonant k state takes part in the polarization. The k-state range that gives V, depends on the occupation factor of fr,k - fl,k. In the case of j C . k- h , k = 0, the range is approximately Ik - k'l < l/Aex, where Aex is the exciton radius. As the occupation factor increases, the range decreases. When the population inversion is achieved with L..k - fL.k > 0, the polarization concentrates on the resonant k state and V, approaches zero, since the Coulomb interaction changes the sign and becomes repulsive, as seen in Eq. (1.245). However, V' still survive, under this condition due to the states with A , k - f;.,k < 0 in the higher energy region. This is the origin of the Coulomb, or excitonic, enhancement of gain. Then, from Eqs. (1.245) and (1.248), the pair function

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MITSURUSUGAWARA

is written as

Using the imaginary part of the polarization derived from Eqs. (1.247) and (1.249), the optical gain becomes

where the broadening function is given as

and Sk(w) represents the variation of amplitude due to the Coulomb interaction. When fE,k fh,k = 0, Sk(w)at w = ee,k+ ejt,k is the Sommerfeld factor. Equation (1.250)is reduced to Eq. (1.42) by setting V, = 0, Sk(w) = 1, and = w , , ~ Comparing . Eqs. (1.251) and (1.42), the influence of the Coulomb effect on optical gain is to change the absolute value above the band edge and to cause the red shift due to the screened exchange and Coulomb-hole effects. Homogeneous broadening of optical gain also occurs due to carrier-carrier Coulomb scattering (see Chapter 7 and the latest news chapter). The above discussion is schematically depicted in Fig. 1.16. Whether the system absorbs light or generates gain is simply determined by the population term of , h , k - f i , k . Equations (1.250) and (1.251) tell us that, when fL.k < 0, an optical absorption resonance corresponding to the excitons appears at an energy of around ( h , k - f i , k ) V l below the band edge. The term of - ( L . k - A , k ) v l corresponds to the exciton binding energy, which decreases as the carrier population increases and vanishes at fi.k - f i , k = 0. At the same time, the band-gap renormalization proceeds, and gain appears in the exciton tail when the population is positive. No gain appears below the band edge under the population inversion of . f r , k - L,,k > 0, since (fc,k - J,k)T/1 > 0. In this sense, bound-state excitons are not formed under the population inversion and never contribute to lasing. What is the advantage of Coulomb gain enhancement on wide-gap laser performance? The most remarkable is that the enhancement might reduce the number of multiple quantum-well layers necessary for lasing. Wide-gap materials like InGaN have large-hole effective mass and thus low-hole

+

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 83 Binding energy = ( f v - fC),EB

't

I

-

CI

0

Q =I

t:

I

" Boundkate" exciton

E

.-0

/ '

4-

Renormalization

a 0

n .E

d

r...\i Unbound stale

m

-’4

Energy

FIG. 1.16. Schematic of optical absorption and gain spectra under the Coulomb interaction.

mobility. This material character gives rise to inhomogeneous hole distribution between quantum wells, such as high-hole concentration in quantum wells in the vicinity of the p-n junction and low-hole concentration in quantum wells far from the p-n junction. As a result, the total material gain is lowered at a given injected current, since quantum wells in the far position work as light absorbers in an extreme case due to low-hole density. If the Coulomb enhancement works well and only one or two quantum-well layers are enough for the lasing threshold, high performance of wide-gap lasers will result.

V.

Quantum-Dot Lasers

Quantum dots have long been expected to drastically improve the performance of semiconductor lasers because their delta-function-like density of states due to three-dimensional carrier confinement enhances gain and differential gain, suppresses the thermal distribution of carriers, gives a nearly zero alpha parameter at the peak gain, and so forth. However, since their potential was first predicted theoretically (Arakawa and Sakaki, 1982) more than ten years ago, quantum-dot lasers have been merely a dream primarily because their fabrication is so difficult. This problem has now been

84

MITSURUSUCAWARA

overcome by the self-assembling process during highly strained epitaxy. In addition to the InGaAs quantum dots treated in this book, a wide variety of others consisting of various semiconductor materials have been fabricated through self-assembling processes, such as InGaP (Carlsson et al., 1995; Reaves et al., 1995), CdSe (Arita et al., 1997), and G a N (Tanaka et al., 1996, 1997). Now, room-temperature lasing from quantum-dot quantized levels is viable, and the threshold current has decreased comparably to that of strained quantum-well lasers (see Chapters 6 and 8). In the near future, this new material category is expected to improve semiconductor laser performance in many aspects, such as threshold current, external quantum efficiency, temperature stability, modulation speed and spectrum linewidth, and in many wavelength regions ranging from infrared to red to blue. Three physically and practically important aspects of quantum dots were not noted in the earlier discussion on quantum-dot laser performance. They are the speed of carrier relaxation into quantum dots, homogeneous broadening of the optical gain spectra of single dots, and the Coulomb effect acting on electrons and holes localized in quantum dots. Retarded carrier relaxation appears to be an inherent problem with the discrete energy levels of quantum dots, since the transition between the discrete levels is significantly slowed due the lack of phonons needed to satisfy the energy conservation rule. This is the so-called phonon bottleneck problem (see Chapter 5). Though this was at one time merely a theoretical problem (Bockelmann and Bastard, 1990 Benisty et al., 1991; Inoshita and Sakaki, 1992), the recent advent of self-assembled quantum dots has enabled us to evaluate carrier relaxation lifetime experimentally. Retarded carrier relaxation, with a lifetime ranging from several tens to hundreds of picoseconds, has been observed in many experiments using time-resolved photoluminescence, some of which confirm the importance of phonon resonant excitation conditions to accelerate relaxation (Mukai et al., 1996a and b; Raymond et al., 1995, 1996). Recent improved theoretical calculations are rather optimistic (Bockelmann and Egeler, 1992; Efros et al., 1995; Gerard, 1995; Nakayama and Arakawa, 1994). For example, an Auger-like process has been suggested for possible fast relaxation channels with a lifetime of ten picoseconds or less. Such fast relaxation is reported to manifest itself in the quick rise of time-resolved photoluminescence and in the initial stage of carrier relaxation, where a large number of carriers surround quantum dots (Ohnesorge et al., 1996; Grosse et al., 1997; Bockelmann et al., 1997). Although more detailed theoretical and experimental work is needed to clarify the relaxation mechanisms, we are sure that the photon bottleneck problem does exist in the sense that the relaxation in quantum dots is slower than in quantum wells, with a typical intraband relaxation lifetime of 0.1 to 1 ps (Asada, 1989).

1 OPTICAL PROPERTIES OF SEMICOND~JCTOR QUANTUM NANO-STRUCTURES 85

A study of single dot optical properties has been one of the most exciting subjects in the field of semiconductor quantum-dot research since the advent of nanocrystals embedded in glass dielectric matrices (Ekimov et al., 1985) and self-assembled quantum dots on semiconductor substrates. A key to observing single dot light emission is to reduce the spatial resolution to about 1 pm or less using, for example, a microscope objective and also to reduce the number of dots in the observation area to less than, say, a hundred. Various unique nature of semiconductor quantum dots has been revealed; a very narrow natural emission linewidth less than 0.1 meV at low temperatures (Zrenner et al., 1994; Brunner et al., 1994; Grundmann et al., 1995; Gammon et al., 1996; Empedocles et al., 1996; Marzin et al., 1997), the broadening and subsequent extinction of the emission line as temperature increases due to homogeneous broadening (Ohta et al., 1997), an intermittent photoluminescence like random telegram signal explained as a result of carrier transfer between a single dot and a deep trap state (Nirmal et al., 1996; Efros et al., 1997). Do these single-dot properties have any correlation with the performance of quantum-dot lasers? If the area density of selfassembled dots is 5 x 10" cm in the active layer of a cavity laser with a cavity length of 300pm and a stripe width of 3pm, the number of dots amounts to 450,000. This dot ensemble have the fluctuation of size and/or semiconductor compositions, and thus, the fluctuation in the resonance energy, which is called inhomogeneous broadening. Because of this large number of hundreds of thousands in the cavity and the energy fluctuation, people would think that single dot optical properties are masked in the operation of conventional cavity lasers. It is pointed out here and in the latest news chapter of this volume that lasing emission spectra are greatly influenced by the magnitude of homogeneous broadening of optical gain of single dots, and that the laser performance is never a mere ensemble of optical properties of single dots. Also unique in quantum dots is the Coulomb interaction among a limited number of carriers confined in quantum dots. The quantum-dot discrete state contains up to two electrons and two holes, and each quantum dot in the ensemble has a different photo-emission energy depending on the number of carriers it contains. The ground state of the filled two-electron and two-hole state with opposite spin directions has lower energy than the one-electron and one-hole states have, as shown both theoretically (Hu, 1990, 1996) and experimentally (Kamada et al., 1997), and can be regarded as a bi-exciton state. Bi-exciton lasing was observed in CuCl quantum dots (Masumoto et al., 1993). This section describes how the carrier relaxation lifetime, homogeneous broadening of the optical gain of single dots, and the bi-exciton effect manifest themselves in quantum-dot laser operation. First, carrier-photon ~

86

MITSURUSUGAWARA

rate equations for quantum-dot lasers are derived to simulate the relationship of output power versus injected current and the small-signal modulation response. Criteria for the carrier relaxation lifetime, as well as inhomogeneous broadening linewidth, dot density and crystal quality, are clarified in how they achieve low-t hreshold, high-efficiency, high-power, and high-speed operation. It is shown that the carrier relaxation lifetime forms a hierarchy in quantum-dot laser performance. Then, the optical gain formula taking into account both inhomogeneous broadening and homogeneous broadening is derived. How lasing emission spectrum varies depending on the magnitude of homogeneous broadening is discussed. After discussing a framework of bi-exciton lasing, we develop an optical gain formula for the bi-exciton-exciton transition and modify the rate equations to describe bi-exciton lasing in quantum dots. Also, we simulate the spontaneous emission and lasing properties of bi-excitons and we discuss the effect of carrier relaxation rate and oscillator strength enhancement on laser operations.

OF CARRIER RELAXATION DYNAMICS ON LASER 1. THE EFFECT PERFORMANCE

Figure 1.17 is an energy diagram of the laser active region, including the self-assembled quantum dots of Fig. l.l(b), and the relaxation process of carriers into the quantum-dot ground state. It is assumed that only a single, discrete electron and hole ground state is formed inside a quantum dot, and that a charge neutrality always holds in each dot. The energy axis represents the energy difference between the conduction-band and valence-band edges. Holes and electrons are treated together, which is enough to check how the retardation of carrier relaxation affects laser performance. The injected carriers diffuse through the separate confinement heterostructure (SCH) layer, relax into the quantum well, and then relax into the dots. Some carriers recombine radiatively or nonradiatively both outside (in the quantum-well region) and inside the dots. Above the lasing threshold, carriers in the ground state emit photons into the lasing mode primarily due to the stimulated emission process. The associated time constants are diffusion in the SCH region (TJ, carrier recombination in the SCH region (TJ, carrier emission from the quantum well to the SCH region (T+J, carrier emission from the quantum dot to the quantum well (re),carrier recombination in the quantum well (T& carrier relaxation into the quantum dot (T~), and recombination in the quantum dot (TJ Needless to say, the discussion below also holds for SK dots accompanied by much thinner wetting layers.

1

OPTICAL PROPERTIES OF

SEMICONDUCTOR

QUANTUM NANO-STRUCTURES

87

FIG. 1.17. Energy diagram of the laser active region, including the self-assembled quantum dots of Fig. l.l(b), and the relaxation process of carriers into the quantum-dot ground state (from Sugawara et al., 1997~).

The rate equations for the carrier-photon system are dN,ldt

=Ile -

dN,/dt

=

NJ?, - N,/?,,

+ NqlTqe

N , / T , -k N I T , - N,/T,,

-

N,/T,,

(1.252) -

N,/T,

(1.253)

and

-

SIT,

(1.255)

where N , is the carrier number in the SCH layer, N , is that in the quantum-well layer, N is that in the quantum dots, V , is the quantum-well layer volume or active-layer volume, and other parameters are in common with those in Section 111.3. The spontaneous emission factor is neglected here. From Eq. (1.49), assuming that Gaussian inhomogeneous broadening is dominant due to the size fluctuation of quantum dots, and substituting the Gaussian function for the Lorentzian function of Eq. (1.43), the

88

MITSURUSUGAWARA

maximum optical gain at the center of the broadening function is given as (1.256) where To is the FWHM of the Gaussian function. In Eq. (1.256), the coverage of dots, 5, is related to the dot density, N , , and dot volume, V,, as

< = N,VD

(1.257)

Under the charge neutrality in each single quantum dot, we set f, = P and f, = 1 - P. The carrier occupation probability, P , is determined by the balance among the rates of carrier relaxation, carrier emission, and photon emission. According to Pauli's exclusion principle, P is related to N as P

=

N/(2NDI/,)

(1.258)

Since carrier relaxation is prevented by the state filling according to the exclusion principle, the relaxation rate is written as 's;

=

(1 - P)ZO1

(1.259)

where r o 1 is the relaxation rate when the ground state is unoccupied, that is, P = 0. From Eqs. (1.56) through (1.58), the third-order optical gain in quantum dots is written as

where (1.261) is the normalized broadening function. Substituting a Gaussian function for Eq. (1.261), the maximum optical gain, including nonlinear susceptibility, is written as (1.262)

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 89

This form is often used in laser simulation to avoid negative gain. The third-order nonlinear coefficient is (1.263) where

rlT = ryl- + l-1-

I.

The relationship of ( 1.264)

is used. Using a set of the above formulae and setting the time-dependent terms of Eqs. (1.252) through (1.255) to be zero, we can calculate steady-state laser operation as a function of the injected current. For simplicity, carrier emission from the quantum dots is ignored; that is, Z/r >> N/T,. This assumption will hold when the quantum-dot energy state is deep enough compared to the thermal energy state. The parameters used are as follows. The dots have a cylindrical shape with a radius of 7 nm and an equal height, ho,,. = 1 eV, considering InGaAs quantum dots; three dot layers are grown; the total optical confinement factor is r = 4%, R , = 30%, R , = 90%0, M = 5 cm-', and L,, = 300ym; and the stripe width is 2pm. The photon lifetime of the cavity is z P = 4.4 ps. The threshold gain is 670 cm from Eq. (1.83). The spontaneous emission lifetime in quantum dots is calculated to be z S p= 2.8 ns using Eq. (1.210) with the exciton effect neglected. The time constants for nonlinear gain are re;' = 100 fs and rli'= z P . Figure 1.18(a-c) shows the output power as a function of the injected current using T~ as a parameter. The spectrum broadening is To = 20 meV, and the coverage is 4 = 10%. which gives P = 0.84 at the lasing threshold and T~ = 6.37, from Eq. (1.259). The recombination lifetimes are (a) T, = T~~ = 2.8 ns and T,, = 3 ns; (b) T , = 2.8 ns and zqr = 0.5 ns; and (c) T , = 0.5 ns , 0.5 ns. Case (a) corresponds to high quality both in the quantum and T ~ = dots and the surrounding quantum well; case (b) corresponds to highquality quantum dots and a low-quality quantum well; and case (c) corresponds to low quality both in the quantum well and in the quantum dots. The lowest threshold current is 110 pA in case (a) with t o = 1 ps. An increase in injected current increases the carrier number in the quantum well, N , , and carriers that relax into the quantum-dot ground state at a rate of N,/rd are transformed into lasing-mode photons through the stimulated emission process. The calculation is stopped when the quantum-well carrier density exceeds 1 x 1012cm-2, because lasing from the quantum well

'

90

MITSURUSUGAWARA 4

F .53

z8 2

CI

P,

0

300 Ps

fg,=

3.0ns

0

(a)

Injected current (mA)

2

4

(b)

Injected current (mA)

6

Injected current (mA)

FIG.1.18. (a)-(c) Output power as a function of the injected current using T~ as a parameter. The spectrum broadening is r, = 20 meV, and the coverage is 5- = 10%. which gives P = 0.84 at the lasing threshold and 7d = 6.35, from Eq. (1.259). The recombination lifetimes are (a) T, = T~~~~ = 2.8 ns and z g r = 3 ns; (b) T~ = 2.8 ns and T~~ = 0.5 ns; and (c) T , = 0.5 ns and tq,= 0.5 ns (from Sugawara et al.. 1998). ( d ) Maximum output power as a function of 70 for various quantum-dot coverages at rt,= 20 meV (from Sugawara et al., 1998).

occurs around this point and this clumps the carrier number in the quantum well, N , , leading to saturation of the output power from the quantum-dot ground state. (We have found experimentally that lasing from the ground state saturates at high output power and then gradually disappears as the lasing from the upper levels starts and its output power increases. The quenching of lasing might be due to the elevated temperature of quantum dots under high current injection and the resultant increase in the emission rate, T ~ . )What is remarkable is as follows: 1. As the relaxation lifetime, to.increases, the threshold current increases

1

OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 91

and the external quantum efficiency, represented by the slope of the curve, decreases. This is because many injected carriers are consumed in the quantum-well region and thus d o not contribute to lasing oscillation. 2. The zo dependence becomes more remarkable when the recombination lifetime in the quantum well, zqrr decreases, as shown in Fig. l.l8(b). The zqr dependence, and thus the quantum-well crystal-quality dependence, of the threshold current and the external quantum efficiency is due to the retarded carrier relaxation, which increases the opportunity for carriers to recombine through the nonradiative process outside the dots. 3. Possible maximum output power decreases as the relaxation lifetime increases. For example, when zo = 1OOps in Fig. 1.18(a), the quantumwell carrier density reaches up to 1 x 1 0 ' 2 c m ~ 2at the power of 3.4mW, causing lasing emission from the quantum well. This is because, as the cavity photon number increases, the extraction of carriers from the quantum-dot ground state accelerates via the stimulated emission process with a rate of S/z,, requiring the increase in N , to supply carriers to the ground state. Since z, is given by Eq. (1.259), the relaxation lifetime can be reduced by lowering P at lasing. Figure 1.18(d) shows maximum output power as a function of T,, for various levels of quantum-dot coverage at To = 20meV. The increase in coverage leads to a lower P , resulting in high output power. If we need a power of 40 mW, we must have t o= 22 ps for 4 = 40% and r0 = 8 ps for 5 = 10%. Note that reducing the linewidth, To, has the same effect on maximum output power as the coverage does, since it increases the optical gain, resulting in lower P at lasing, shorter relaxation lifetime, and higher maximum power. 4. As the recombination lifetime in quantum dots decreases, the threshold current increases correspondingly, as seen in Fig. 1.18(c) due to the consumption of carriers inside the quantum dots. This occurs even for the rapid relaxation case. For a small-signal current injection with a modulation angular frequency of w,, that is, I = I , + I(o,) exp(iw,t), let us check the response of laser output. The response function is defined as M(w,) = eS(wn,)/1(com),where S = So + S(c0,) exp(iqnt). Though the function has a rather complex form, it can be simplified by the following assumptions: (1) z,, << zS; ( 2 ) z, << rqr, zqe; and (3) z, << z,. Each means that (1) carriers survive to reach the quantum well; (2) all carriers in the quantum well relax into quantum dots; and (3) emission of carriers from quantum dots is negligible. These assumptions correspond to high-quality quantum dots with a deep confinement

92

MITSURUSUCAWARA

potential. Also, these are enough for our purpose which is to clarify how the modulation bandwidth is limited by the relaxation lifetime. The resultant response function is given as

Q

1

M(wm)= (1

+ o;tj)l’2(1 +

c1&rf)”2

[(oz, - of)’

+ y om]l / 2 2

2

(1.265) (1.266) (1.267) (1.268)

and (1.269)

where 0,is the relaxation oscillation frequency, y is the damping factor, and g‘ is the differential gain. The major factors that limit modulation bandwidth are relaxation oscillation frequency and rolloff due to retarded carrier relaxation. The parameter that dominates relaxation oscillation frequency is the differential gain and thus the spectrum width. The 3-dB bandwidth limited by the relaxation lifetime of sd is given as (1.270)

Figure 1.19(a) shows the frequency dependence of the modulation response function with parameters of zo = 10 ps, To = 20 meV, and 5 = 10%. The response function shows the resonance at the relaxation oscillation frequency, which increases with the output power. The rolloff of the response due to the retarded relaxation is also seen. In Fig. 1.19(b), the 3-dB bandwidth is plotted as a function of t o for various levels of coverage with the broadening of To = 20 meV. The increase in coverage leads to lower P. For 10-GHz modulation, we need t o = 7ps for = 40% and t o = 3 ps for 5 = 10%. For 40-GHz modulation (not shown in the figure), z, should be close to 1 ps. Requirements for the relaxation lifetime are much more severe in high-speed modulation than in static performance.

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 93 2, 1

m

5 = 10%

( P = 0.84)

1.4 mw\ h

ro=20meV -

3 1.5-.

v

13.8 mW

Frequency, f = d2n: (GHz)

(a)

20

P

15

9 OD

10

c

5

'0

(b)

10

20

30

40

Relaxation lifetime, ro (ps)

50

(c)

Output power (mw)

FIG. 1.l9. ( a ) Frequency dependency of the modulation response function with parameters of T" = lops, To = 20meV, and 5 = 10%. (b) 3dB bandwidth as a function of T" for various coverages. (c) Relaxation oscillation frequency ( 1;.=
Figure 1.19(c) shows the relaxation oscillation frequency (.f, = wr/2.rr) as a function of output power using To as a parameter. The dot coverage is 5 = 10% for To = 5 , 10, and 20meV; 5 = 20% for To = 40meV; and i; = 40% for To = 80 meV. The relaxation oscillation frequency is a good measure of modulation bandwidth as long as the limit imposed by the retardation of carrier relaxation is not taken into account. The linewidth narrowing gives rise to the increase in J.; for example, f , reaches 40GHz when To = 10 meV, and 57 GHz when To = 5 meV at the power of 20 mW. At these narrow linewidths, the alpha parameter comes close to zero, giving an almost negligible wavelength chirp. This gives birth to a directly

94

MITSURUSUGAWARA

modulated high-speed quantum-dot laser as a practical device. Note again that in order to take full advantage of this potential; we must overcome the rolloff of the response arising from retarded carrier relaxation into quantum dots. From what we have seen from the rate equation analyses, there is a hierarchy in quantum-dot lasers that depends on relaxation lifetime (Fig. 1.20). First of all, for sub-mA low-threshold operation, quantum dots should have a narrow spectrum linewidth of 20-30 meV and be of high quality with the carrier lifetime close to the intrinsic spontaneous emission lifetime. The relaxation should be accelerated as we go from sub-mA low-threshold to highly efficient, high-power, high-speed operation. If the carrier relaxation lifetime is reduced to around 10 ps, quantum-dot lasers will be able to show high output power, high external quantum efficiency, and low threshold currents. The slope efficiency of quantum-dot lasers will be superior to that of quantum-well lasers because lower carrier density at lasing leads to low internal loss. Actually, the measured efficiency of our columnar-shaped quantum-dot lasers is excellent (see Chapter 6). If 1.3-pm lasing is achieved in InGaAs/GaAs quantum-dot lasers (see Chapter 3), the temperature characteristics of the threshold current will be greatly improved for the following reason. Conventional strained quantum-

Hierarchy of quantum-dot lasers A

7

Q

u

Direct modulation lasers

2z

i!

- High quality

I

V

QD lasers

I lo-’

I

11111111

1

I

11111111

10

I

11111111

102

I

i l l l d

1o3

Relaxation lifetime (ps)

FIG. 1.20. Hierarchy in quantum-dot lasers, depending on the relaxation lifetime. For sub-mA low-threshold operation, quantum dots should have a narrow spectrum linewidth of 20-30meV and be of high quality with the carrier lifetime close to the intrinsic spontaneous emission lifetime. The relaxation should be accelerated as we go from sub-mA low-threshold to highly efficient, high-power, and high-speed operation. If the carrier relaxation lifetime is reduced to 1 ps and, at the same time. the spectrum linewidth is narrowed to less than IOmeV, quantum-dot lasers will work as high-speed direct-modulation lasers with almost zero-chirp.

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 95

well lasers emitting in the infrared (1.3 and 1.55 pm grown on InP substrates have suffered from the low temperature stability of threshold currents, primarily due to carrier overflow from the active region to the SCH region with relatively low potential height (Ishikawa, 1993; Ishikawa and Suemune, 1994). Carrier overflow into SCH regions with high density of states increases the carrier number at lasing and thus accelerates the Auger recombination process, the rate of which is proportional to the cube of the carrier density, resulting in the consumption of carriers. At the same time, large carrier density at lasing greatly increases the internal cavity loss, resulting in low external quantum efficiency. In self-assembled InGaAs quantum dots grown on GaAs substrates, carrier overflow is greatly reduced, since high-potential barrier (SCH) and clad materials like AlGaInAs and AlGaInP can be grown on GaAs substrates to prevent carrier overflow. As a result, quantum-dot lasers on GaAs substrates, especially infrared lasers for optical transmission systems, will, in many respects, be superior to strained quantum-well lasers on InP substrates. If the carrier relaxation lifetime is reduced to 1 ps and, at the same time, the spectrum linewidth is narrowed to less than 10 meV, quantum-dot lasers can work as high-speed direct-modulation lasers with almost zero chirp. Note that the GHz modulation of strained quantum-well lasers is done by external quantum-well modulators using the quantum-confined Stark effect. This is because, in strained quantum-well lasers. wavelength chirp is an inevitable problem for high-speed modulation. A high-speed direct-modulation laser, which was predicted by Arakawa and Sakaki in 1982, might be one of the goals in quantum-dot laser research. How long is the relaxation lifetime? To what extent is the relaxation accelerated? As stated in the introduction to this section, much more work should be done to find solutions to this problem. In addition to improved theoretical and experimental studies on carrier relaxation itself, detailed experimental analyses of quantum-dot laser performance will give a clear picture of carrier relaxation dynamics. The thinking presented in this section will help analyze laser operations. Finding a way to accelerate relaxation through the design of a quantum-dot active layer might be required. Experimental evaluation of the phonon bottleneck problem is presented in Chapter 5.

2. EFFECTOF HOMOGENEOUS BROADENING OF ON LASINGEMISSION SPECTRA

OPTICAL

GAIN

The formula for the optical gain spectrum is presented here, taking into account both inhomogeneous and homogeneous broadening. Using the

96

MITSURUSUGAWARA

formula, we discuss the lasing emission spectra of quantum-dot lasers. (See the latest news chapter for the experimental results.) The optical gain formula of Eq. (1.49) for quantum dots has homogeneous broadening of Lorentz shape with a width that is proportional to the transverse relaxation rate due to scattering (Bcs in Eq. (1.43)). In the actual quantum-dot lasers, we need to take into account inhomogeneous broadening due to dot size fluctuation and their spatial isolation as

where Ec,, is the average energy, and the FWHM is given as To = 2.35&. The optical gain in the laser cavity is then given by the convolution integral of the two broadening factors as

(1.272) Equation (1.272) indicates that the optical gain at a given energy, E , is created by collecting the individual contribution of all dots within the homogeneous broadening of single dots around that energy. If the homogeneous broadening is a delta function, that is, B,,(E) = 6(E), Eq. ( 1.272) becomes

In this case, dots with different energies have no correlation to each other and gives optical gain independently since they are spatially isolated from each other. As a result, all dots with an optical gain above the lasing threshold start lasing emission independently, which leads to broad band lasing emission as depicted in Fig. 1.21(a). The quantum-dot laser at this situation behaves in the same way as if it included independent lasing media in the same cavity. When homogeneous broadening of the optical gain is comparable to inhomogeneous broadening, lasing-mode photons receive gain not only from energetically resonant dots but also from other nonresonant dots that lie within the amount of homogeneous broadening. Since carriers of nonresonant dots are extracted and put into the central lasing modes by the

1

OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 97 (b) B(E) E G(E)

FIG. 1.21. Relationship between lasing spectra and homogeneous broadening of single-dot optical gain. (a) If the homogeneous broadening is a delta function, all dots with an optical gain above the lasing threshold start lasing emission independently, which leads to broad-band lasing spectrum. (b) When the homogeneous broadening of the optical gain is comparable to the inhomogeneous broadening, carriers of non-resonant dots are extracted and put into the central lasing modes by the stimulated emission. leading to lasing spectrum with one central line.

stimulated emission, lasing emission with one line occurs as depicted in Fig. 1.21(b). Note that the calculation in Sec. V.1 implicitly assumed this collective lasing. See our recent experiments on the lasing emission spectra of our originally-designed columnar shaped quantum dots in the latest news chapter in this volume, where experimental evidence for the above discussion is presented and the origin of homogeneous broadening of optical gain is discussed.

3. BI-EXCITONSPONTANEOUS EMISSIONAND LASING Excitons, in the sense of electron-hole bound states, are not formed under the population inversion bet ween the conduction and valence bands, and cannot generate optical gain, as shown in Section IV.7. The Coulomb interaction affects the optical gain such that the gain’s absolute value is enhanced, a phenomenon introduced earlier Coulomb, or excitonic, enhancement, and such that the gain spectrum is red-shifted due to screened exchange energies and Coulomb-hole self-energies. For bound-state excitons to generate gain, the exciton emission should be shifted toward lower energies from the exciton absorption resonance through some mechanism. This becomes possible by the assistance of third partners besides photons and excitons, such as scattering particles, bi-excitons, or localized states. The bi-exciton-exciton transition process is one of the possible candidates for exciton-related lasing (Shaklee et al., 1971; Leheny and Nahory,

98

MITSURU SUGAWARA

1971; Masumoto et al., 1993; Shionoya, 1995; Kreller et al., 1995; Sugawara, 1996, 1997b, 1998; Kozlov et al., 1996). A bi-exciton, analogous to a hydrogen molecule, is formed through the coupling of a pair of excitons. The ground-state bi-exciton is composed of two excitons with opposite spin directions and has a binding energy of B,,, as seen in Figure 1.22(a). The bi-exciton binding energies increase as the ratio of effective mass of m,*/mt increases. For example, the binding energy is calculated to be about 0.07EB at m,*/m: = 0.2 (Akimoto and Hanamura, 1972). The oscillator strength of

Bi-exicton dispersion 4M

2M

Wave number, K

(b) Two excitons Formation ._I.

Exciton

-

Ground state

(c) FIG.1.22. (a) Three-level model of a bi-exciton. an exciton and a ground state. B,, is the bi-exciton binding energy. (b) Energy-wave vector dispersion relationships of excitons with a wave vector of K, and bi-excitons with wave vector of K,. (c) Bi-exciton formation process. Excitons excited by pumping, whether electronically or optically, collide with each other and form bi-excitons. A bi-exciton annihilates through spontaneous emission and the stimulated emission process, leaving one exciton behind.

1

OPTICAL PROPERTIES OF SEMICONDUCTOR

QUANTUM NANO-STRUCTURES 99

the bi-exciton-exciton transition is proportional to the wave function extent of the bi-exciton and is enhanced from that of the exciton transition. The energy-wave-vector dispersion relationships of excitons with a wave vector of K, and bi-excitons with a wave vector of K, are shown in Fig. 1.22(b), assuming that the dispersions are isotropic with the translational mass of M,* for excitons and M:, = 2M,* for bi-excitons. A bi-exciton annihilates by emitting a photon, leaving one exciton behind. The optical transition occurs almost perpendicularly in the energy-wave-vector space with the selection rule of K, = K, q 3 K, = K. This rule is similar to the ordinary electron-hole interband optical transition. The emitted photon has less energy than does the exciton optical absorption resonance by B,, h2K2/(4Mz).In the extreme, a combination of one bi-exciton and no exciton generates optical gain for the electromagnetic field with the energy of hwbe. This is in contrast to the optical gain due to ordinary electron-hole transition, where many carriers are needed to achieve population inversion. In this sense, a bi-exciton is analogous to a single quantum dot filled with two electrons and two holes in the ground states. Thus, a key to bi-exciton lasing is whether bi-excitons outnumber excitons to generate gain equal to the threshold gain of a cavity, and whether this occurs at lower carrier densities than the exciton and bi-exciton Mott densities. For these conditions to be achieved, bi-excitons should be more stable than excitons, and the bi-exciton formation should occur faster than the bi-exciton and exciton annihilation through the spontaneous and stimulated emission processes. The expected large oscillator strength of bi-exciton-exciton transitions help to lower the threshold carrier density. Figure 1.22(c) illustrates the bi-exciton formation process. Excitons excited by pumping, whether electronically or optically, collide with each other and form bi-excitons. A bi-exciton annihilates through the spontaneous emission and stimulated emission processes, leaving one exciton behind. The corresponding rate equations for the bi-exciton number, N,,, and the exciton number, N , , are

+

+

for bi-excitons, and dN,ldt

=

G

-

N,/z,

+ 2N,,/z,, + 2(N,,

-

N:,)/r,

+ (c/n,)g,TS

(1.275)

for excitons, where z, is the bi-exciton spontaneous emission lifetime. r , is the exciton spontaneous emission lifetime, z, is the interconversion time between bi-excitons and excitons, and G is the pumping rate. Nonradiative recombination is not considered here. The photon rate equation in a cavity

100

MITSURt!

SUGAWARA

is similar to Eq. (1.255), with the nonlinear gain effect omitted for simplicity. The bi-exciton gain is given as

where gb is the differential gain. The bi-exciton number under the thermal equilibrium, N:!, is given by the Saha equation (Kim et al., 1994) as N2.y = Ni/N*

(1.277)

with (1.278) for quantum wells. The spontaneous emission can be modeled by setting S = 0 in Eqs. (1.274) and (1.275). For the steady-state solution, the spontaneous emission intensity is given as

Ixx =-2Nxx

=

GJ(1

+ G/G,)”’

- 13’

(1.279)

‘xx

for bi-excitons and (1.280) for excitons, where (1.281) When G << Go, the bi-exciton emission intensity is proportional to G 2 , while the exciton emission intensity is proportional to G. As G increases, the biexciton intensity exceeds the exciton intensity at G = 8Go. The system reaches transparency at G = GJ(1 + 4tx/zx,)’ - 11’. Figure 1.23 shows the total exciton density-2Nx, + N,-needed to reach transparency ( N , = N x x ) as a function of temperature. The lifetimes are set as z,, = 45 ps and z, = 60 ps. At low temperatures, where the thermal energy is about one order of magnitude smaller than the bi-excion binding energy, bi-excitons

I

OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 101

c

.-00

lO O

CI

X

-Q 12 c1

lo8 10’

0

50

100

150

200

250

300

Temperature (K) density. ( N , + 2N,,)/D, needed to

FIG. 1.23. Total exciton reach transparency as a function of temperature, using B,, and T~ as parameters. The horizontal dashed line represents the exciton saturation density due to the phase-space-filling efiect. (Reprinted from Sugawara,
are stable and population inversion is realized at quite low densities. The temperature range where the electron-hole system stabilizes as bi-excitons is 10-20 K for B,, = 10 meV and below 30 to 50 K for 30 meV. The above scenario of bi-exciton lasing seems unlikely to occur in wide-gap semiconductors. One reason is that the bi-exciton binding energy is around lOmeV, or 30meV at most, in 11-VI and InGaN materials, requiring low temperatures for bi-exciton stability. The other is that spatial energy fluctuations due to composition fluctuations lower the optical gain at a low injection density, since only a small fraction of the active region volume can contribute to the gain. This leads to electron-hole plasma lasing with the local potential minima fully filled with a large number of carriers. Similarly, quantum dots with large-size inhomogeneity cannot generate enough optical gain for lasing, so lasing caused by electron-hole plasma occurs from the quantum well or wetting layers. As long as spatial energy fluctuations are not controlled to give homogeneous samples, bi-exciton lasing is not observed.

102

MlTSURU SUGAWARA

If we fabricate well-controlled homogeneous local potential minima as quantum dots, we have a chance. Each discrete level of quantum dots incorporates up to two electrons and two holes with opposite spin directions according to the Pauli exclusion principle. The filled state of two electrons and two holes state in the quantum-dot ground state has a positive binding energy close to the exciton binding energy in bulk materials (Hu, 1990; Hu et al., 1996), and thus this two-pair state can be called the bi-exciton ground state. The emission spectrum caused by the bi-exciton-exciton transition appears red-shifted from the emission of a one-electron-hole state, that is, the exciton state. In blue-regime semiconductor quantum dots, the biexciton binding energy should reach several tens of meV, and we expect the bi-exciton emission to be well resolved from the exciton emission in spite of inhomogeneous broadening. What is interesting about bi-excitons in quantum dots is that we can study various aspects of bi-exciton lasing by artificially designing quantumdot structures. Let us examine a one-to-one correspondence between quantum wells and quantum dots:

1. In quantum wells, the bi-exciton and exciton excited states and many exciton states appear along with the ground-state excitons. By controlling the size and strains of quantum dots and the band-gap discontinuity against surrounding materials, we can design the number of discrete states in the dots. If there is only one in both the conduction and valence bands, a pure bi-exciton state is formed when the state is fully filled with two electrons and two holes. If quantum dots contain more than two levels, the bi-exciton and exciton-excited states and many exciton states appear, depending on the number of carriers in the dot. 2. For bi-exciton lasing, bi-excitons should be more stable than excitons, and the bi-exciton formation should occur faster than the bi-exciton and exciton annihilation. The bi-exciton stability in quantum dots can be improved by increasing the quantum-dot potential barrier height. The counterpart of the bi-exciton formation rate in quantum dots is the relaxation rate discussed in Section V.1. The rate is controlled to some extent by designing the quantum-dot energy levels and by devising injection methods. 3. The Mott density of bi-excitons is a matter to be investigated. We can experimentally study how the carrier number around quantum dots (in the quantum well in the model of Section V.1) affects the dots’ binding energy and oscillator strength, by changing the dot density, the laser output power, etc. To take into account the bi-exciton effect in quantum dots, let us use the

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 103

three-level model, where the bi-exciton and exciton states are taken as base functions. The bi-exciton and exciton system is described by the Hamiltonian of (Hu, 1990; Hu et al., 1996)

where hwy is the ground-state energy, h o e is the exciton energy, hw, is the bi-exciton energy, lg) is the ground state, le) is the exciton state, and Ib) is the bi-exciton state. The transition between the eigenstates of the system occurs as a result of the interaction of the electromagnetic field with the wave vector, q, and the polarization, 0. The interaction with a field is described as

where p,,(t) is given by Eq. (1.176) and p h e ( t ) is given similarly. The procedure to derive the optical gain is the same as in Section 111; that is, the density-matrix equation is solved with the initial condition of pg’+ C, pi:’+ C,&) = 1. The polarization of the system is given from the diagonal summation of the density matrix-polarization product. As a result, the linear optical gain is given as

where weY= me - cog, whe= w b - w, = weg- B J h . The broadening functions, Be, and Bhe, are the Lorentz functions with a full width at half maximum of dephasing rate 2Feg and 2rb,, respectively. The first term of Eq. (1.284) is the result of the exciton-ground-state transition, where the oscillator strength is given by Eq. (1.172) as (1.285) Here, $c(r, R) is the exciton envelope wave function. The second term of Eq. (1.284) represents the bi-exciton-exciton transition, where the oscillator

104

MITSURUSUGAWARA

strength is given as

where $be(rl, R,, rz, R,) is the bi-exciton envelope wave function and the subscripts of 1 and 2 represent the first and second excitons. Qualitatively, Eq. (1.286) shows that the oscillator strength of the bi-exciton-exciton transition is proportional to the product of the absolute square of the first-exciton overlap integral, the probability of finding an electron and a hole at the same position in the second exciton, and the volume covered by the second exciton. The oscillator strengths of Eqs. (1.285)and (1.286)can be measured as a ratio against the oscillator strength for the electron-hole transition of

(1.287) where a,, is the interband optical transition frequency at the band edge. The oscillator strength, f: and ,j":, was calculated by Takagahara (1989) as a function of the radius of the spherical quantum dot. It increases as the dot radius increases, a phenomenon known as mesoscopic enhancement. According to Takagahara's calculation, j;,/fo2 4, when the radius is equal to the bulk exciton Bohr radius. To calculate the population balance between ground, exciton, and biexciton states, let us use the occupation probability, P, defined by Eq. (1.258). The probability of finding a bi-exciton in quantum dots is P 2 , the probability for a spin-up exciton is P(l - P), the probability for a spindown exciton is P(l - P), and the probability of not finding any excitons is (1 - P ) 2 . Thus, we obtain pi:)

-

p::,'

= P2

-

P(l - P)= 2p2 - p

(1.288)

and pL0,' - p g ) = P(l

-

P ) - (1 - P),

= -1

+ 3P

-

2Pz

(1.289)

The population inversion appears when P > lj2 for both transitions. Bi-exciton lasing from quantum dots can be described using rate equations only slightly different from those in Section V.l in the following points. Considering quantum dots that include only the ground-state exciton and

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 105

bi-exciton levels, Eq. (1.284) gives the maximum optical gain at the biexciton-exciton transition as (1.290) Again, the Gaussian broadening function arising from the size inhomogeneity of quantum dots is assumed to be dominant. The spontaneous emission rate is given as N

-

=2N,Vu

rr

LXX

1

2P(1 - P)

-P+2

z,

where

(1.291)

( 1.292)

for bi-excitons ( i = xx) and excitons ( i = x), and the emitted photon frequency is (0. The relaxation rate of Eq. (1.259) is unchanged, since ~d-'

=(1

-

+

P ) 2 ~ i 12P(1 - P)(tG1/2) = (1 - P ) T ~ ' (1.293)

Setting S = 0 for the spontaneous emission process and omitting the time-dependent terms in Eqs. (1.252) through (1.254), we obtain the current dependence of P and the spontaneous emission intensity as

I,

=

2NDP(1 - P )

(1.294)

5,

for excitons and (1.295) for bi-excitons. The spontaneous emission and steady-state laser output power as a function of current can be calculated by the same procedure as in Section V.l. The parameters considering wide-gap InGaN quantum dots are as follows. The dots have a cylindrical shape with a radius of 3.6 nm (equal to the exciton Bohr radius in bulk materials) and an equal height; 5 = 5%, hw,,. = 3.0 eV, To = 20 meV, and L,, = 300 pm; three dot layers are grown; the total optical confinement factor is = 4%; R , = 30%, R , = 90%, and a = 5 cm-'; and the stripe width is 2pm. Since the bi-exciton binding

106

MITSURUSUGAWARA

energy is estimated to be 30meV in a quantum dot of this size, the bi-exciton emission is well separated frrom the exciton emission. Figure 1.24(a) shows the spontaneous emission intensity, I , and I,,, as a = 4, function of the injected currents. The oscillator strength is set at f,,/fo and the spontaneous emission lifetime is calculated to be T,, = 0.34 ns by Eq. (1.292). The relaxation rate is set at lops. (We have no idea whether this is an realistic value. To realize this fast relaxation for quantum dots, the tuning of energy levels to LO phonon energy might be needed.) The bi-exciton emission intensity increases in proportion to the square of the current, while the exciton emission increases in proportion to the current. The bi-exciton emission exceeds the exciton emission intensity at the current of 1.1mA. As seen in the inset of the figure, the lasing threshold is 1.4mA. Figure 1.24(b) shows the maximum output power and the occupation, P , at the lasing threshold as a function of oscillator strength. Again, the “maximum” is defined as the output power when the carrier density around the dots reaches 1 x 1012cm-2. As the oscillator strength increases, P decreases, increasing the maximum output power for the same physical reason as in Section V . l . This increase is the most beneficial point of the bi-exciton effect, or Coulomb enhancement, of the optical gain in quantum dots. Note that the oscillator strength formulae of Eqs. (1.285) and (1.286) correspond to the case in which there are no carrier number in the quantum well surrounding the quantum dots, N , , increases as the laser output power increases. This decreases the bi-exciton binding energy and the oscillator strength enhancement. At the same time, the homogeneous broadening of

B P

2

3

I

4

Oscillator strength, fxx/fo Current (mA) Fiti. 1.24. (a) Spontaneous emission intensity due to excitons, I , . and bi-excitons, I,,, in the quantum dot laser as a function of injected currents. The bi-exciton emission intensity exceeds the exciton emission at a current of I.lmA. The lasing threshold is 1.4mA (from Sugawara et al., 1998). (b) Maximum output power and the occupation, P , at the lasing threshold as a function of oscillator strength (from Sugawara et al., 1998). (a)

1 OPTICAL PROPERTIES OF SEM~CONDUCTOR QUANTUM NANO-STRUCTURES 107

the optical gain will increase because of scattering, resulting in smaller optical gain, When the carrier density reaches the level at which population inversion occurs in the quantum-well regime, the bi-exciton binding energy and the oscillator strength enhancement will have almost disappeared, and lasing will occur from the quantum-well regime. From this, we see that bi-exciton lasing properties, such as emission energy, optical gain, and output power, are dominated by the carrier relaxation and emission rate, and the energy level structures of quantum dots.

VI. Summary This chapter provided a theoretical background of the optical properties of semiconductor quantum nano-structures, emphasizing how those properties vary from quantum wells to quantum dots and how they are influenced by the exciton effect. We saw that optical gain, optical absorption, and spontaneous emission depend strongly on the dimension of quantum nano-structures and on the exciton effect. The importance of accelerating carrier relaxation into quantum dots in the realization of high-performance quantum-dot lasers was stressed. A framework of bi-exciton lasing was discussed. The following physical properties of quantum dots remain to be clarified: (1) the relaxation rate into quantum dots by electron-electron and electron-phonon scattering and by tunneling injection; (2) the laser structure design to realize fast relaxation, ( 3 ) the dephasing time of the carrier-photon interaction via carrier-carrier and carrier-phonon scattering to determine the linear and nonlinear susceptibility spectra, that is, homogeneous broadening, (4) the many-body effect of a limited number of electrons and holes in quantum dots to clarify the Coulomb effect on the optical gain spectrum; and (5) the band engineering of quantum dots and surrounding materials for fast carrier relaxation and strong carrier-photon interaction. These issues, which are now being studied by many researchers, will help us to create new high-performance quantum-dot optical devices.

Appendix

To determine optical transition strength, the matrix element of

108

Mr-rsuuu SUGAWARA

in Eq. (1.20) should be calculated. For the interband transition between the conduction and valence bands, using Eq. (1.21), we get

The matrix element of (c, k,le'q''e,

- p1u, k,) is evaluated as follows.

Bulk Muteriuls From Eqs. (1.2) and (1.4), the electron wave function in the conduction band is

-

ik, r)u,(r) and that in the valence band is

where u,(r) is for the conduction band and u,(r) is for valence bands. Thus, the matrix element is given as (c, k,le'q"e,~ pJu,k,) =

J +

exp[i(k, - k,,

$1,

exp[i(k,

+ q)

+ q)

*

r]d3r

Jn

u,*(r)e, pu,(r)d3r

-

rle, * p exp[ik, r]d3r

u,*(r)u,,(r)d3r

(A.4)

The integration is separated into the product of the integration over the unit cell and over the entire crystal because the envelope function varies slowly over the unit cell. Since the Bloch states in the conduction band and the

1

OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 109

valence band are orthogonal,

where the wave vector selection rule is

Thus, Eq. (1.31) for the interband transition is written as PPr,k= (c, kle, * plu, k )

=

u,*(r)e, * pu,,(r)d3r

(A.7)

Quantum Wells From Eqs. (1.2) and (1.7), the electron wave function in the conduction band is

and the valence-band state is

The matrix element is

I

(c, k,le'q''e,- plu, k,) = d k ~ l , ~ ~ l ~ q l I ~ n , , . f l ,u:(r)e;pu,,(r)d3r ,

(A.lO)

where

(A.ll) is the overlap integral between the quantized envelope wave function of the conduction and valence bands. Here, since the well-layer thickness is much smaller than the wavelength of the electromagnetic field, the electric dipole approximation of e'Yz"= 1 is used. For the infinite potential well, the

110

Mr~suuuSUCAWARA

selection rule for the quantum number in the z direction is

In actual quantum wells with a finite potential height, Z,lcz,na,2 is less than 1 even when ncz = nrz. The wave vector selection rule is

kII = kII

- qII

-

kII 3

(A.13)

Thus, Eq. (1.31) for the interband transition is written as P

Quantum Dots

Since quantum dots are much smaller than the photon wavelength, the electric dipole approximation of e'q" = 1 holds. Thus,

(A.15) The next problem is to calculate jnu;(r)e; pu,(r)d3r. According to Kane's k * p method of calculating the band dispersion around the band edges, the periodic part of the band-edge Bloch functions is taken as (Kane, 1956, 1957)

r

for uc, where and J are spin functions and S has atomic s-orbital symmetry. The functions for the valence band, u , , are

for the heavy-hole valence band,

(A.18)

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTUKES 111

for the light-hole valence band, and

(A.19) for the split-off band resulting from the spin-orbit interaction, where X , X and Z have p-orbital symmetry. The quantization axis of angular momentum is taken in the z direction. The P-state bases of Eqs. (A.17) through (A.19), written in ( J , M , ) representation, are taken to diagonalize the spin-orbit interaction. The electron wave vector is assumed in the z direction, and the base functions rotate with k vectors. We should first calculate the matrix element with the z-axis at the k direction and then transform the element into the xyz coordinate system. Finally, we average the matrix elements over degenerate k vectors depending on the symmetry of the crystal. If the system has structural asymmetry, as do quantum wells and quantum wires, the matrix element has polarization and k-vector dependence, according to the valence-band type related to the optical transition.

Bulk Materials The momentum matrix element is obtained by averaging Eq. (A.7) with respect to the solid angle. Then.

(P$k12= I(Slpli)12/6 s M 2 (i

=X

, X and 2 )

(A.20)

The k *p perturbation gives the transition matrix in bulk materials as

(A.21) where mo is the free electron mass, 15, is the band gap, A is the spin-orbit splitting energy, rn: is the conduction-band-edge effective mass, and D' is the second-order perturbation term (Sugawara, 1993b).

Quantum Wells The wave vector is (k",n,n/L,). Let 6' be the angle between the z axis and the wave vector. The character gives the polarization dependence to the

112

MITSURUSUGAWARA

matrix element as (Yamanishi and Suemune, 1984) (A.22) where M&4ell,

k) = 3(1

+ cos2H)/4

(A.23)

and

M&(eL,k)

=

3 sin2H/2

(A.24)

for conduction to heavy-hole band transition, M&(ell, k) = (1

+ cos28)/4 + sin2@

(A.25)

and

M&(el, k) = sin28/2 + 2 cos2H

(A.26)

for conduction to light-hole band transition, and (A.27) for conduction to split-off-band transition. Quuntum Dots The averaging procedure of matrix elements is done over degenerate k states, depending on the symmetry. If a quantum dot has a spherical shape, the averaging gives

IP&12 = M2

(A.28)

REFERENCES Akimoto, 0.and Hanamura, E. (1972). J. Phys. Sot.. Jpr7. 33, 1537. Andreani, L. C., Tassone, F., and Bassani, F. (1991). Solid Sfcite Commun. 77, 641. Arakawa, Y., and Sakaki, H. (1982). Appl. Phys. Left. 40, 939. Arita, M., Avramescu, A,, Uesugi, K., Suemune, I.. Numai, T.. Machida, H., and Shimoyama, N. (1997). J p n . J . Appl. P h ~ x36. 4097. Asada, M. (1989). IEEE J . Quantum Electron. 25, 2019.

1 OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 113 Bastard, G., Brum, J. A,, and Ferreira. R. (1991). “Electronic States in Semiconductor Heterostructures,” in Solid State Physics 44, Academic Press, New York. Benisty, H., Sotomayor-Torres, C. M.. and Weisbuch, C. (1991). Phvs. Reu. B. 44, 10945. Bockelmann, U., and Bastard, G. (1990). Phys. Rev. B. 42, 8947. Bockelmann, U., and Egeler, T. (1992). Phys. Reu. B. 46, 15574. Bockelmann, U., Heller, W., Filoramo, A., and Roussignol, P. (1997). Phys. Rev. B. 55, 4456. Brunner, K., Abstreiter, G., Bohm, G . , Trinkle, G., and Weimann, G. (1994). Phys. Rev. Lett. 73, 1138. Cao, H., Jacobson, J., Bjork, G., Pau, S., and Yamamoto, Y. (1995). Appl. Phys. Lett. 66. 1107. Carlsson, N., Georgsson, K., Montelius. L.. Samuelson, L., Seifert, W., and Wallenberg, R. (1995). J . Cryst. Growth 156, 23. Chang, L. L., Esaki, L., and Tsu, R. (1974). A p p l . Phys. Lett. 24, 593. Chemla, D. S., Damen, T. C., Miller, D. A. B., Gossard, A. C., and Wiegmann, W. (1983). Appl. Phys. Lett. 42, 864. Chemla, D. S., Miller, D. A. B., Smith. P. W., Gossard, A. C., and Wiegmann, W. (1984). I E E E J . Quantum Electron. QE-20, 265. Chow, W. W., Wright, A. F., and Nelson, J. S. (1996). Appl. Pliys. Lett. 68, 296. Chuang, S. L. (1991). P ~ J J Reo. S . B. 43, 9649. Chuang, S. L. (1995). Physics ufOptoelec,fronirsDeoices. John Wiley & Sons, New York. Citrin, D . S. (1993). Phys. R ~ I JB.. 47, 3832. Ding, J., Jeon, H., Ishihara, T., Hagerott, M., Nurmiko, A. V., Luo, H., Samarth, N., and Furdyna, J. (1992). Phj’s. Reu. Left. 1707. Ding, J., Hagerott, M.. Ishihara, T., Jeon, H., and Nurmiko, A. V. (1993). Phjx Rev. B. 47. 10528. Dingle, R. (1975). Festkorperprobleme 15, 21. Efros, A. L., and Rosen, M. (1997). Phys. Rer. Lett. 78, 1110. Efros. A. L., Kharchenko, V. A,, and Rosen. M. (1995). Solid State Cornmun. 93, 281. Ekimov, A. L., Efros, A. L., and Onushchenko, A. A. (1985). Solid State Commun. 56. 921. Elliot, R. J. (1957). Phys. Rev. B. 108, 1384. Empedocles, S. A., Norris, D. J., and Bawendi. M. G. (1996). Phys. Rec. Lett. 77. 3873. Esaki, L., and Tsu, R. (1969). IBM Resetrrch Note. RC-2418. Esaki, L., and TSU,R. (1970). I B M J . Res. Deiwlop. 14, 61. Feldmann, J.. Peter, C.. Gobel, E. 0..Dawson. P., Moore, K., Foxon, C., and Elliott, R. (1987). J . P h w . Reo. Lett. 20, 2337. Galbraith, I. ( 1995). “Excitonic gain processes in ZnSe,” in Microscopic Theorj of Semiconductors: Qumtum Kinetics. Confinemen/ rind Ltrsers, World Scientific Publishing Co. Pte. Ltd., Singapore. Gammon, D., Snow, E. S., Shanabrook, B. V., Katzer, D. S.. and Park, D. (1996). Phys. Rev. Lett. 76, 3005. Gerard, J.-M. (1995). “Prospect of high efficiency quantum boxes obtained by direct epitaxial growth,” in Confined Electrons mid Photons: New Physics and Applicutions. Plenum. New York. Goldstein, E. V.. and Meystre, P. (1995). ”Spontaneous Emission and Laser Oscillation in Microcavities.” CRC Press, Boca Raton. FL. Grosse, S.. Sandmann. J. H. H. (1997). Phys. Ret:. B. 55, 4473. Grundmann, M., Christen, J., Ledentsov. N. N., Bohrer, J.. Bimberg. D.. Ruvimov. S. S., Werner, P.. Richter, U.. Gosele. U., Heydenreich, J., Ustinov, V. M.. Egorov, A. Yu, Zhukov, A. E., Kop’ev, P. S.. and Alferov, Zh. I. (1995). PIiys. Reu. Lett. 74, 4043. Haase, M. A. (1991). Appl. Phys. Lett. 59. 1272. Haken, H. (1973). Quuntenfeldtheorie drs Fesfkurpers. B.G. Teubner, Stuttgart.

114

MITSURUSUCAWARA

Hanamura, E. (1988). Phys. Rec. B. 38, 1228. Haug, H., and Koch, S. W. (1977). Phys. Stilt. S O / . 82, 531. Haug, H., and Koch, S. (1994). Quuntum Theory of the Optical und Electronic Properties qf Semiconductors, 3rd ed. World Scientific, Singapore. Hayashi, I., Panish, M. B., Foy, P. W., and Sumski, S. (1970). Appl. Phys. Lett. 17, 109. Houdre, R., Stanley, R. P., Oesterle. U., Ilegems, M., and Weisbuch, C. (1994). Phys. Rev. B. 49, 16761. Hu, Y. Z., Lindberg, M., and Koch. S. W. (1990). Phys. Rev. B. 42, 1713. Hu, Y. Z., Gieben. H., Peyghambarian, N., and Koch, S . W. (1996). Phys. Rev. B. 53, 4814. Inoshita, T., and Sakaki. H. (1992). Phys. Rev. 5. 46, 7260. Ishibashi, T., Tarucha, S., and Okamoto. H. (1981). Proc. lnr. Symp. GaAs und Related Compounds, Oslo. Ishikawa, H. (1993). Appl. Pkys. Leu. 63. 712. Ishikawa, H., and Suemune, I. (1994). Photon Tech. Lett. 6, 344. Itoh, T. (1992). Nonlineur Optics 1, 61. Jacobson. J. et al. (1995). Phys. Reu. A. 51. 2542. Jen, Y., Tsutsumi. T., Souma. I., Oka. Y., and Fujiya. H. (1993). Jpn. J . Appl. Phys. 32, L1542. Kadoya, Y., Kameda, K., Yamanishi. M., Nishikawa, T., Kannari, T., Ishihara, T., and Ogura, 1. (1995). Appl. Phys. Lett. 68, 281. Kamada, H., Ando, H.. Temmyo, J., and Tamamuira, T. (1997). Exr. Abs. 58th Autumn Meet. Jupun Society of Applied Physics 0, Akita. Kane, E. 0. (1956). J . P h j ~Chern. Solids 1. 82. Kane, E. 0. (1957). J . Phys. Chem Solids 1. 249. Kawakami, Y., Hauksson, I., Simpson, J., Stewart, H., Galbraith, I., Prior, K. A,, and Cavenett, B. C. (1994). J . Cryst. Growth 138. 759. Kim, J. C., Wake, D. R., and Wolfe, J. P. (1994). Phys. Rev. B. 50, 15099. Klingshirn, C., and Haug, H. (1981). Phys. Reu. 5. 70, 315. Knox, R. S. (1963). Theory ofExcitons. Academic Press, New York. Koch, S. W., Haug, H.. Schmieder, G.. Bohnert, W., and Klingshirn, C. (1978). Phps. Srut. Sol. 89, 431. Kozlov, V. et al. (1996). Phys. Rev. 5. 53, 10837. Kuroda, Y., Suemune, I., Fuji, Y., and Fujirnoto, M. (1992). Appl. Phys. Lett. 61, 1182. Kreller, F.. Lowisch, M., Puls, J., and Henneberger, F. (1995). Phys. Rev. Lett. 75, 2420. Leheny, R. F., and Nahory, R. E. (1971). Phjs. Rec. Lett. 26, 888. Luttinger, J. M., and Kohn, W. (1955). Phys. Rev. B. 97, 869. Kuwatsuka, H., and Ishikawa. H. (1994). P h w . Rev. B. 50. 5323. Marzin, J.-Y., Gerard, J.-M., Izrael, A.. Barrier. D., and Bastard, G. (1997). Plij~.Rec. Lett. 73, 716. Masumoto, Y., Kawamura, T., and Era, K. (1993). Appl. Phys. Lett. 62, 225. Meystre, P., and Sargent, M. (1990). Elements of’ Quuntuni Optics. Springer-Verlag, Berlin. Miller, D. A. B., Chemla, D. S., Smith, P. W.. Gossard, A. C., and Tsang, W. T. (1982a). Appl. Phys. Rev. B. 28, 96. Miller, D. A. B., Chemla, D. S., Eilenberger. D. J.. Smith. P. W., Gossard, A. C., and Tsang, W.T. (1982b). Appl. Phys. Leu. 41, 679. Miller, D. A. B., Chemla, D. S., Eilenberger, D. J., Smith, P. W., Gossard, A. C., and Wiegmann, W. (1983). Appl. Phys. Lett. 42, 925. Miller. D. A. B., Chemla, D. S., Damen. T. C.. Gossard, A. C., Wiegmann, W., Wood, T. H., and Burrus. C. A. (1984a). Phys. Rec. Lett. 53, 2173. Miller, D. A. B.. Chemla, D. S., Damen, 7.C.. Gossard, A. C., Wiegmann, W., Wood, T. H., and Burrus, C. A. (l984b). Appl. Phps. Lerr. 45, 163.

1

OPTICAL PROPERTIES OF SEMICONDUCTOR QUANTUM NANO-STRUCTURES 115

Miller, D. A. B., and Burrus, C. A. (1985a). Phys. Rev. B. 32, 1043. Miller, D. A. B., Chemla, D. S., Damen, T. C., Wood, T. H., Burrus, C. A., Gossard, A. C., and Wiegmann, W. (1985b). IEEE J . Quuntutn Electron QE-21, 1462. Mukai, K., Ohtsuka, N., Shoji, H., and Sugawara, M. (1996a). Appi. Pkys. Lerr. 68, 3013. Mukai, K., Ohtsuka, N., Shoji, H., and Sugawara, M. (1996b). Phys. Re[).B. 54, R5243. Nakamura, S., Senoh, M., Nagahama, S., Iwasa, N., Yamada, T., Matsushita. T., Kiyoku, H., and Sugimoto, Y. (1996). Jpn. J . Appl. Phys. 35, L74. Nakayama, H., and Arakawa, Y. ( 1 994). 7th Int. Conf Superluttices, Microstructures, and Microdevices, Banff, Canada. Nirmal, M., Dabbousi, B. O., Bawendi, M. G., Macklin, J. J., Trautman, J. K., Harris, T. D., and Brus, L. E. (1996). Nature 383, 802. Norris, T. B., Rhee, J. K., Sung, C. Y., Arakawa, Y., Nishioka, M., and Weisbuch, C. (1994). Phys. Rev. B. 50, 14663. Ogawa, T. (1995). “Dimensionality and Optical Responses of Materials,” in Opricul Properties of Low-Dimensional Muteriuls. World Scientific, Singapore. Ohnesorge, B., Albrecht, M., Oshinowo, J., Forchel, A,, and Arakawa. Y. (1996). Phys. Rei). B. 54, 11532. Ohta et al. (1989). Physicu, E2, 573. Raymond, S., Fafard, S., Charbonneau, S., Leon, R., Leonard, D., Petroff, P. M., and Merz, J. L. (1995). Phys. Rev. B. 52, 17238. Raymond, S., Fafard, S., Poole, P. J., Wojs, A,, Hawrylak, P., Charbonneau, S., Leonard, D., Leon, R., Petroff, P. M., and Merz. J. L. (1996). Phys. Rev. B. 54, 11548. Reaves, C. M., Bressler-Hill, V., Varma. S., Weinberg, W. H., and DenBaars, S. P. (1995). Surf: Sci. 326, 209. Schmitt-Rink, S., Chemla, D. S., and Miller, D. A. B. (1989). Adv. Phys. 38, 89. Shaklee, K. L., Leheny, R. F., and Nahory, R. E. (1971). Phys. Rev. Lett. 26, 888. Shinada, M., and Sugano, S. (1966). J . Phys. Soc. J p n . 21, 1936. Shionoya, S. (1995). Kotai-Butsuri 5, 438 [in Japanese]. Sugawara, M., Fuji, T., Yamazaki, S., and Nakajima, K. (1990). Phys. Rev. B. 48. 8102. Sugawara, M. (1992). J . Appl. Phys. 71, 277. Sugawara, M., Okazaki, N., Fuji, T., and Yamazaki, S. (1993a). PAys. Rev. B. 48, 8102. Sugawara, M., Okazaki, N., Fujii, T.. and Yamazaki, S. (1993b). Pkys. Reu. B. 48, 8848. Sugawara, M., and Yamazaki, S. (1994). Microiouve und Optic. Tech. Lett., 7, 107 (special issue). Sugawara, M. (1995). Phys. Rev. B. 51, 10743. Sugawara, M. (1996). Jpn. J . Appl. Phy.s. 35, 124. Sugawara, M. ( 1997a). Proc. 2nd Int. Cnnf. Nitride Semiconductors, Tokushima. Sugawara, M. (1997b).Jpn. J . Appl. P h w . 36, 2151. Sugawara, M., Mukai, K., and Shoji, H. (1997~).Appl. P h j ~ sLett. . 71, 2791. Sugawara, M. (1998). Proc. Optoelectronics Y8,Photonic West, San Jose, CA. Takagahara, T. (1989). Phys. Rev. B. 39, 10206. Tanaka, S., Iwai, S., and Aoyagi, Y. (1996). Appl. Phys. Lett. 69. 4096. Tanaka, S., Hirdyama, H., Ramvall, P.. and Aoyagi, Y. (1997). Proc. 2nd Int. Con/. Nitride Senziconrluctors, Tokushima. Thijs, P. J. A., and van Dongen, T. (1990). E.ut. Ahs. 22nd Int. Cur$ Solid Siute Devices mid Muteriuls, Sendai, Japan. Tsang, W. T. (1981). Appi. Phys. Lett. 39. 786. Uenoyama, T. (1995). Phys. Rev. B. 51, 10228. Weisbuch, C. (1987). Semiconductors trnd Srmirnetals, 24, Academic Press, New York. Weisbuch, C., Nishioka, M., Ishikawa, A., and Arakawa, Y. (1992). Phys. Rer. Lett. 69, 3314. Weisskopf, V., and Wigner, E. P. (1930). 2. Phys. 63, 54.

116

MITSURUSUGAWARA

Wood, T. H., Burrus, C. A,, Miller, D. A. B.. Chemla, D. S., Damen, T. C., Gossard, A. C., and Wiegmann, W. (1984). Appl. Phys. Lett. 44, 16. Yamanishi, M., and Suemune, I. (1984). J p n . J . Appl. Phys. 23, L35. Yokoyama, H., and Ujihara, K. (1995). "Spontaneous Emission and Laser Oscillation in Microcavities." CRC Press, Boca Raton, FL. Zhang, 2. L., Nishioka, M., Weisbuch. C., and Arakawa, Y . (1994). Appl. Phys. Lett. 64, 1068. Zrenner, A,, Butov, L. V., Hagn, M. Abstreiter, G.. Bohm, G., and Weimann, G . (1994). Phys. Rev. Lett. 12, 3382.

SEMICONDUCTORS A N D SEMIMETALS. VOL. 60

CHAPTER 2

Molecular Beam Epitaxial Growth of Self-Assembled InAs/GaAs Quantum Dots Yoshiaki Nakata and Yoshihiro Sugiyama QUANTUM ELECTRON DEVICES LABORATllRY FUIITSULABORATORIES LTD JAPAN ATSUGI.KANACAWA.

Mitsuvu Sugawara OPTIC'AL SEMICONOUCTOR DEVICES

LABOKATORY

FUIITSULABORATORIES LID ATSUGI,KANAGAWA, JAPAN

I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . 117 THESTRANSKI-KRASTANOW GROWTH MODE. . . . . . . . . . . . . . . . 119 1. Energy- Balance Model,for Islcind Formation . . . . . . . . . . . . . . 119 2. InAs Island Growfh . . . . . . . . . . . . . . . . . . . . . . . . . 121 3. Multiple-Layer Growth anti Perpendicular Alignment of Islands . . . . . . 125 4. In-Plane Alignment of Ishncis . . . . . . . . . . . . . . . . . . . . 130 111. CLOSELY STACKED InAsJGaAs QUANTUM DOTS. . . . . . . . . . _ . _ _ 132 1. Close Stacking of InAs Islunt6 . . . . . . . . . . . . . . . . . . . . 133 2. Photoluminescence Properties . . . . . . . . . . . . . . . . . . . . 137 3. Zero-Dimensional Exciton Corlfinenwnt Evaluated by Diamagnetic Shifts. 140 I V . COLUMNAR InAsJGaAs QUANTUM DOTS. . . . . . . . . . . . . . . . . 143 v. S U M M A R Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I52 11.

1

I. Introduction The atom-like density of states in quantum dots should drastically improve the performance of optical devices, especially semiconductor lasers, and should also be instrumental in the development of novel optoelectronic and single-electron devices. Since high-performance lasers, including quantum-wire and quantum-dot in active regions, were first proposed theoretically by Arakawa and Sakaki (1982), fabricating these low-dimensional 117 Copyright 1 1999 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-752169-0 ISSN 0080-8784/99 $3000

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YOSHIAKI NAKATA AND YOSHIHIROSUGIYAMA

quantum nano-structures has been a major interest in semiconductor research. In particular, 111-V nano-structures such as GaAs/AlGaAs, InGaAs/GaAs, and InGaAs/InP are major targets in optical and electronic device applications. Numerous challenges in quantum-wire and quantum-dot fabrication have been reported during the last two decades. The most straightforward technique is to laterally pattern the quantum-well structures through a combination of high-resolution electron beam lithography and dry or wet etching (Petroff et al., 1982; Reed et al., 1986; Miyamoto et al., 1987; Gershoni et al., 1988). Other techniques exploit regrowth of epitaxial layers, such as fractional layer growth on a vicinal substrate (Petroff et al., 1984; Fukui and Saito, 1987), selective growth on a patterned substrate (Kapon et al., 1989, 1992), and cleaved-edge overgrowth (Goni et al., 1992; Pfeiffer et al., 1993; Wegscheider et al., 1995). However, artificial structures fabricated in these ways did not take full advantage of engineered energy states, and some had drawbacks. For example, lithography- and etching-based technologies caused damage to the crystals, such as impurity contamination, defect formation, and poor interface quality. Further, there were serious problems in the fabricated structures themselves, such as large size, low density, and size irregularity. Significant quantum-size effects advantageous to, say, optical devices appear when the size is less than the exciton Bohr radius. Moreover, millions of the structures should be densely packed, with size uniformity on the atomic scale, to obtain the desired optical signal. Unfortunately, no technique could succeed in achieving all of these requirements. Self-assembling, a novel way to fabricate quantum dots, is now being welcomed as the most promising approach in overcoming the various problems with previous techniques. This process exploits the three-dimensional island growth of highly lattice-mismatched semiconductors. The growth of InAs on a GaAs substrate is a typical example, where the lattice mismatch between InAs and GaAs is about 7%. Dislocation-free highdensity coherent islands of InAs are self-assembled on the GaAs substrate, accompanied by a wetting layer. Typical InAs self-assembled islands have a dome or pyramid shape with a base length of about 20 nm and a height of a few nm. Since the exciton Bohr radius is 10 to 20nm in an InAs-GaAs system, the island size is small enough to exhibit the three-dimensional quantum confinement effect. Actually, well-separated photo-emission spectra from discrete energy states have been observed (Mukai et al., 1994; Grundmann et al., 1996). Room-temperature lasing from the quantum-dot quantized levels has been achieved, and the threshold current has decreased and is now close to that of strained quantum-well lasers (see Chapter 6). Though self-assembling is a new way to form quantum dots, the growth

2

MOLECULAR BEAMEPITAXIAL GROWTH

119

process itself is not new but is simply a Stranski-Krastanow (SK) mode growth (Stranski and Krastanow, 1937). InGaAs/GaAs islands grown via the SK mode were observed and evaluated by several groups in the mid-1980s (Schaffer et al., 1983; Lewis et al., 1984; Glas et al., 1987; Houzay et al., 1987). However, the research did not attract broad interest, at least as it applied to quantum dots. That SK InAs islands on a GaAs substrate might work as quantum dots was proposed by Tabuchi et al. (1992). Since then many works have identified the islands as quantum dots with threedimensional quantum confinement, primarily on the basis of their optical emission properties (Leonard et al., 1993; Mukai et al., 1994; Marzin et al., 1994; Oshinowo et al., 1994; Moison et al., 1994). Three-dimensional island growth, now the center of interest in semiconductor material and device research, is known as self-assembled growth. In addition to the InGaAs quantum dots treated in this volume, a variety of dots of other semiconductor materials have been fabricated through self-assembling processes. These include InGaP (Carlsson et al., 1995; Reaves et al., 1995), CdSe (Arita et al., 1997), and GaN (Tanaka et al., 1996, 1997). In the near future, this new material category will improve various optical devices like semiconductor lasers in many aspects (see Chapter 7) and in wavelength regions ranging from infrared to red to blue. This chapter focuses on the molecular beam epitaxial (MBE) growth of InAs islands on a GaAs substrate. Section 11, briefly reviews the three well-known growth modes and a simple energy balance model for island formation, and then describes the growth process of InAs three-dimensional islands with various diagnostic results. Then, Section I11 introduces the perpendicular stacking of InAs islands, called closely stacked quantum dots, and Section IV introduces columnar-shaped quantum dots. Perpendicular stacking increases the island size primarily in the growth direction and enables us not only to tune the emission wavelength but to narrow the spectrum linewidth caused by the island-size fluctuations. By comparing the structural and optical properties of different types of dots, we can see the perpendicularly stacked columnar shaped quantum dots’ advantages. 11. The Stranski-Krastanow Growth Mode

1. ENERGY-BALANCE MODELFOR ISLAND FORMATION The crystal growth on a bulk substrate occurs in one of three distinct modes, schematically shown in Fig. 2.1; Frank-van der Merwe (FM), Volmer-Weber (VW), and Stranski-Krastanow (SK). While growth proceeds layer by layer in the FM mode, VW growth causes three-dimensional

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YOSHIAKINAKATA AND YOSHIHIRO SUGIYAMA

Frank-van der Merwe mode

Volmer-We&r mode Island

Epitaxial layer

FIG. 2.1. Three types of growth mode on a substrate; Frank-van der Merwe (FM); Volmer-Weber (VW); and Stranski-Krastanow (SK). In the F M mode growth proceeds layer by layer. In the VW mode growth occurs as three-dimensional islands on the substrate. The SK mode is a combination of the FM and VW modes, where the growth of a severalmonolayer-thin film, called a wetting layer. is followed by cluster nucleation, leading to island formation.

islands on the substrate, if the film has a higher surface energy than that of the substrate. SK growth is a combination of the FM and VW modes, where the growth of a several-monolayer thin film, called a wetting layer, is followed by cluster nucleation and then to island formation. Which growth mode occurs depends primarily on the difference in the surface energy between the substrate and the grown material and on the strain energy accumulated in the grown materials as a result of lattice mismatch. In the strained-layer growth of semiconductors discussed here, the SK growth mode has been found to dominate. Why islands form in the lattice-mismatched epitaxy can be understood by a simple model of energy balance (Tu et al., 1992). Let us compare the total energy between a coherently strained film on a substrate and strain-free islands made by the same number of atoms as in the film. The total energy is supposed to consist of the strain energy due to lattice mismatch and the surface energy. The strain-free islands are, for simplicity, assumed to be cubic shaped with a side length of X . Instead of releasing the strain energy, the cubic islands have greater surface energy due to the increased surface area. The island energy is lower than the film energy, then, when the island is larger than a critical value determined by the surface energy, y , and the in-plane strain, E, as

x > x,K

y/&2

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When E # 0, there is always a value of X above which the island is in the lower energy state; that is, when the side length of the islands is greater than X,, the island configuration will be favored. This indicates that, given sufficient growth temperature and time, islands are always formed in the strained layer epitaxy. In contrast, the present strained-layer epitaxial growth using metalorganic chemical vapor deposition (MOCVD) and MBE enables us to obtain coherent strained layers over 1% lattice mismatch without dislocations, which tells us that strained-layer epitaxy occurs under metastable conditions, and that surface kinetics also play an important role. Though the present energy balance model looks too simple, in Eq. (2.1) we see that the critical size is inversely proportional to the square of the strain and that we will have smaller islands as the lattice mismatch increases. In the following sections, we will see that this simple rule explains actual island growth rather well.

2. InAs ISLAND GROWTH

When InAs is supplied to a GaAs substrate, three-dimensional island growth occurs followed by the two-dimensional growth of a wetting layer. Before we move on to the islands’ perpendicular stacking, let us see in detail how the growth of individual islands proceeds. With MBE growth, the reflection high-energy electron diffraction (RHEED) pattern gives us important information on the surface state. When the layer-by-layer growth proceeds, RHEED shows streak patterns. The RHEED pattern transits to spots when three-dimensional islands start to grow. For the post-growth surface morphology evaluation technique, we use atomic force microscopy (AFM), from which we know the shape, size, density, alignment, and the distance between neighboring islands. We can get similar information from plan-view transmission electron microscopy (TEM) even when islands are covered by overgrown layers, we can also evaluate defects, such as dislocations and stacking faults. Cross-sectional TEM is quite useful for observing cross-sectional shape and height of the islands, and how the islands are stacked perpendicularly and interact with each other. We will see this in Sections I11 and IV. For the growth of InAs islands, a conventional MBE was used with metallic In, Ga, Al, and As, as source materials. The substrates were (001)-oriented GaAs. Before growth, they were thermally cleaned at about 680°C for one minute under an arsenic pressure of 1.2 x lo-’ Torr. InAs islands were grown on a GaAs (100 nm)/AlGaAs (50 nm)/GaAs (400 nm) buffer layer and covered with a GaAs (50 nm)/AlGaAs (50 nm)/GaAs (100 nm) cap layer. The substrate temperature for the growth of InAs was

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varied between 470 and 560°C, and was 650°C for the growth of the buffer layer. The substrate temperature was monitored by a pyrometer calibrated by the melting temperature of In-A1 alloys. The growth rates of InAs and GaAs were approximately 0.1 and 0.75 pm/h. In other words, it takes about 10 seconds to grow one monolayer (ML) InAs and about 1.3 seconds to grow one ML GaAs. The arsenic Torr and was 1.2 x pressure used for InAs island growth was 6 x Torr for buffer-layer growth. These values were measured by an ion gauge at the substrate position. Two arsenic cells were used, both set at 6 x lop6 Torr. The arsenic pressure was changed abruptly at the interface by switching one of these cells on and off. After 60-second annealing of the InAs islands, GaAs layers were overgrown. The variation of the RHEED pattern and the diffraction intensity are shown in Fig. 2.2. During the growth of the buffer layer, the surface was

Growth time (s)

2 0 ~ 1 div. 1

FIG.2.2. Variation of the RHEED pattern and the reflection intensity for the various stages of growth of lnAs islands. The pattern transition from streaks to spots began as the growth of lnAs reached around 1.6 monolayers at 16 seconds, indicating that two-dimensional growth transited to three-dimensional island growth. The reflection intensity reached an almost constant value at the 1.8-ML InAs supply. The pattern went back to streaky at the 6-ML GaAs growth, showing that 6-ML GaAs completely covers the InAs islands and almost flattens the surface.

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As-stabilized (2 x 4) and then transited to c(4 x 4) reconstruction while the substrate temperature dropped to 510°C. The leftmost photograph in Fig. 2.2 is the [ l l O ] azimuthal RHEED pattern obtained before InAs growth. The lp-order fractional diffractions were clearly observed. Immediately after the InAs growth started, the fractional-order reflections disappeared, and the transition from streaks to spots started as the growth of InAs reached around 1.6 M L (at 16 seconds), indicating that a two-dimensional layer growth transited to a three-dimensional island growth. When the GaAs supply started to overgrow the InAs islands as a cap layer, the spot intensity rapidly decreased. The pattern went back to streaky at the 6-ML GaAs growth, showing that the 6-ML GaAs supply fully covers the InAs islands and almost flattens the surface. The island size and density were evaluated by ex-situ AFM operated in the air. Figure 2.3 shows AFM images of the island surfaces grown with the InAs supply of 1.3, 1.6, 2.1, and 2.6 ML. These images refer to different epilayers grown under the same conditions. The scanned area is 0.5 x 0.5 pm2. The island growth started at an InAs supply of about 1.6 ML, as seen in the change of RHEED pattern. As the amount of InAs supply

(c) 2.1 ML

(d) 2.6 ML

2 0.5 x 0.5 pm FIG. 2.3. AFM images of the islanding surfaces when the InAs supply was 1.3, 1.6, 2.1, and 2.6 MLs. The islanding growth starts at the 1.6-ML supply, as seen in the change in RHEED pattern. As the supply amount increases, the dot density rapidly increases and the dots coalesce.

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increased, the island density rapidly increased, and the islands began to coalesce when the amount of InAs exceeded approximately 2.3 ML. Figure 2.4 shows the in-plane base diameter and the density of the islands as a function of the amount of InAs supply (InAs nominal thickness (ML)). As depicted in the figure, growth proceeds from the two-dimensional growth of a wetting layer to the nucleation of InAs islands at about a 1.6-ML supply, to the increase in the island density, and finally to the coalescence of neighboring islands at about a 2.3-ML supply. Figure 2.5 shows the base diameter and density of islands grown with the 2.1-ML InAs supply as a function of the growth temperature. The AFM images of islands grown at each temperature are also shown. As temperature increases from 470 to 56OoC,the island diameter increases from 15 to 45 nm and the density decreases correspondingly. This temperature dependence is due to the surface migration lengh of TnAs, which increases as the temperature increases. Figure 2.6 shows the plan view (a) and the (1 10) cross section (b) of TEM images for InAs islands grown with a 2.5-ML InAs supply. Observable were

lnAs nominal thickness (ML) FIG. 2.4. In-plane diameter and density of islands as a function of the InAs nominal thickness. The growth proceeds from the two-dimensional growth of a wetting layer, to the nucleation of InAs islands at about 1.6 ML supply. to the increase in the density, and finally, to the coalescence of neighboring islands at 2.2-2.3-ML supply.

-

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60

lo'*

Ts = 510°C

540°C

0 450 500 550 600 Growth temperature ( "C)

560°C 0.5 x 0.5 pm2

FIG. 2.5. Diameter and density of islands at 2.1-ML InAs supply as a function of growth temperature, and the AFM images of islands at each temperature. As the temperature increases from 470 to 560'C. the island diameter increased from 15 to 45 nm and the density decreases correspondingly.

two types of islands-small, with a diameter of 15-20 nm, and large, with about twice the diameter -which were formed probably by coalescence. While the height of most islands is 3 to 5 nm, that of the large-size islands exceeds 10nm. Contrasts due to strain were observed both in the GaAs buffer layer and in the GaAs overlayer, and they extended as the dot size increased. There are two types of defects: V-shaped dislocations and multiple stacking faults. In the upper middele part of Fig. 2.6(a), a large island accompanied by line-shaped defects on both sides could be seen. These defects are a V-shaped dislocation pair lying on two equivalent { 11l } planes. Note that both types of defect are generated from large islands and almost no defects are observed around the small islands (Ueda et al., 1998).

3. MULTIPLE-LAYER GROWTH AND PERPENDICULAR ALIGNMENT OF ISLANDS The InAs islands buried in GaAs work as quantum dots because they are smaller than the exciton wave function both in the in-plane and perpendicular directions. When we apply the TnAs island layers to the active region of

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FIG. 2.6. (a) the (001) plan-view and (b) the (110) cross-sectional TEM images of InAs islands (2.5-ML InAs coverage). Observed are small islands with a diameter of about 20nm and large islands with about twice that diameter formed by the coalescence of two islands. While the height of most islands is 3 to 5 nm, that of the large islands exceeds 10 nm. Contrasts due to strain are observed in both the GaAs buffer layer and the GaAs overlayer, and they extend as the dot size increases. There are two types of defect to be observed: V-shaped dislocations and multiple-stacking faults.

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optical devices, multiple-layer stacking of the island layer is often required to obtained enough interaction between confined electrons and the electromagnetic field. This is immediately understood if we think of multiple quantum-well lasers. Figure 2.7 shows the cross-sectional TEM images of the stacked island structures grown with the intermediate layer thickness of lOnm (a) and 20nm (b). For the multiple-layer stacking, the growth sequence shown in Fig. 2.2 is simply repeated. These intermediate-layer thicknesses are much greater than the island height of about 3-5 nm. In the 20-nm structure (b), the islands form independently of the lower-layer islands. AFM images show that the mean island size and the density are almost constant in all island layers. When the intermediate layer is reduced to 10 nm, the islands align in the perpendicular direction and the size increases as we move to the upper layers. This perpendicular alignment of islands is due to the strain fields induced by the lower layer islands. Xie (1995) provided an analytical description of correlated island formation in the growth direction under strain fields. His explanation follows: Islands on the first layer produce tensile stress in the GaAs intermediate layer above the islands. Let 1, be the effective length of the lateral strain field extent at the GaAs surface. The InAs that forms the next layer impinging inside I, around the lower island preferentially migrates from regions of higher to lower lattice mismatch and accumulates just on top of the lower islands. Thus, the InAs can achieve an energetically lower thermodynamic state due to lower lattice mismatch against the GaAs in tension. The strain fields induced by the islands provide the driving force for

(a) Intermediate layer: 10 nm

(b) Intermediate layer: 20 nm

FIG. 2.7. Cross-sectional TEM images taken when the intermediate GaAs layer thickness was (a) 10 nm and (b) 20 nrn. These intermediate layer thicknesses are greater than the height of the InAs islands, which is 3-5 nm. In the 20-nm structure, the island formation occurs independently between layers. When the intermediate layer was reduced to 10 nm, the islands aligned in the perpendicular direction and the size increased as growth moved on to the upper layers.

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perpendicularly aligned self-assembling island formation. In the region outside l,, where there is negligible stress, some of the impinging In atoms initiate formation of islands. Thus, in order to achieve effective perpendicular alignment, the island separation, I, should satisfy the condition of 21, > 1. This condition is achieved by decreasing the intermediate GaAs thickness below a critical intermediate layer thickness. The size and density of islands vary as the multiple-layer stacking proceeds, as seen in Fig. 2.8. When we grew 1.8-ML InAs islands using 10-nm thick intermediate GaAs layers, the average diameter of the 10thlayer islands was about 45 nm, 90% larger than the first-layer islands, which were 24 nm in diameter. The island density decreased from 1 x 10" cm-2 for the first-layer islands to 3 x 10'ocm-2 for the 10th-layer islands. These phenomena were also reported by Solomon et al. (1996). The increase in the island size and the decrease in the density can be understood by the above-mentioned strain field model as follows. The size distribution of self-assembled islands leads to spatial fluctuation of strain fields at the surface of GaAs. Supplied InAs for the growth of the next island layer preferentially accumulates on the sites with larger strain fields. Due to smaller lattice mismatch between InAs and the surface, the island becomes large. As a result, large islands are formed preferentially on the large-strainfield region, preventing island formation on the small-strain-field region. This explains the increase in the island size and the decrease in density in multiple-island growth with about a 10-nm intermediate layer. As seen in Fig. 2.8, as we further reduce the thickness of the intermediate layer below 10 nm, the increase in size and the decrease in density become less remarkable. For example, when we grew multiple layers with 3-nm thick intermediate layers, the average diameter of the 10th-layer islands was about 33 nm-40% larger than the 1st-layer islands, with 24-nm diameter- while the islands' density decreased from 1 x 10' cm2 for the 1st-layer islands to 8 x lOl0cm2 for the 10th-layer islands. This is because the thinner intermediate layer causes the strain field over small islands to become large enough to accumulate supplied InAs. Photoluminescence spectra from single-layer and multiple-layer samples with 20-nm intermediate layers at 77 K are shown in Fig. 2.9. The sample was excited by an Ar' ion laser at a power of 1 mW, and the luminescence was dispersed by a monochrometer and detected by a cooled Ge detector. The laser spot was about 100pm in diameter. The emission spectrum appeared at around 1.2 eV with a full width at half maximum (FWHM) of 90meV for both samples. Such large spectrum width is typically observed in self-assembled SK-mode islands so far reported. This is inhomogeneous broadening, caused by the fluctuation of the quantized energies among islands included in the measured area (lo6 to lo7 islands). By means of

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129

Intermediate layer thickness, L (nm)

FIG. 2.8. In-plane diameter and density normalized against the first-island-layer density as a function of the intermediate-layer thickness. When 1.8-ML InAs islands were grown using LO-nm thick intermediate GaAs layers, the average diameter of the 10th-layer islands was about 45m, 90% larger than the first-layer islands, with a 24-nm diameter. The island density decreased from 1 x 10" em-' for the first-layer islands to 3 x 10" cm-' for the 10th-layer islands. As the thickness of the intermediate layer was further reduced below 10 nm, the increase in size and the decrease in density became less remarkable.

microprobe photoluminescence to access a limited number of islands, a sharp emission spectrum with around 100peV is observed (Marzin et al., 1994; Fafard et al., 1994; Grundmann et al., 1995a; Leon et al., 1995; Hessman et al., 1996). When we apply SK-mode islands as the quantum dots to the laser active region, the large spectrum broadening lowers the optical (differential) gain and prevents us from achieving the high performance predicted in quantumdot lasers. For example, low gain leads to lasing from excited levels with a high density of states, increasing the threshold current. Low differential gain lowers the relaxation oscillation frequency and limits the modulation bandwidth. One exceptional device that prefers large broadening is multi-

YOSHIAKINAKATA AND YOSHIHIROSUGIYAMA

PL 77 K Stacked island layers FWHM: 90 meV

0.9

1.0

1.1 1.2 Energy (eV)

1.3

1.4

FIG. 2.9. Photoluminescence spectra from single-layer and multiple-layer samples with 20-nm intermediate layers at 77 K. The emission spectrum appeared at around 1.2 eV with the full width at half maximum (FWHM) of 90meV for both samples. This inhomogeneous broadening was caused by the fluctuation of the quantized energies among islands included in the measured area (lo6 to lo’ islands).

wavelength optical memory, where the spectrum hole burning is exploited for data storage and the large spectrum broadening increases the memory size (Muto, 1995). The broadening of luminescence should be controlled by the structural fluctuations, especially by the island height, since the SK-mode islands have a height of 3-5nm, which is much smaller than the diameter of about 20nm. This is understood by a well-known concept that the quantized energy changes by a constant size fluctuation as the size of the confinement region decreases. Thus, if the height can be increased or more accurately controlled, emission spectrum broadening will be greatly reduced. Increase or control is achieved by means of close stacking of islands in the perpendicular direction, as will be seen in Sections I11 and IV. 4. IN-PLANE ALIGNMENTOF ISLANDS

The islands become technologically more interesting if we can manipulate their arrangement laterally as well as vertically to achieve three-dimensional arrays. There are several reports on lateral ordering, in-plane alignment, and

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control of islanding sites in specific regions, which we summarize in the following paragraphs. The short-range ordering of InAs SK-mode islands grown on (001) GaAs by MBE was reported by Grundmann et al. (1995b), who showed by plan-view TEM observation that InAs islands align in the quasi-periodic square lattice with an axis of (100). Shchukin et al. (1995) theoretically explained this spontaneous ordering by showing that a periodic array of strained islands arranged in a two-dimensional square lattice has a minimum total energy with main axes along the [loo] and [OlO] directions. Tersoff et al. (1996) presented a model showing that island size and spacing grow progressively more uniform as successive island layers are stacked. Nishi et al. (1997) observed spontaneous lateral alignment of InGaAs islands grown by gas-source MBE aligned in a direction inclined about 60" from the [Oll] direction on (311)B surfaces. Surface steps play an important role in determining strained-island nucleation. Leonard et al. (1994) and Ikoma and Ohkouchi (1995) showed by AFM and ultra-high-vacuum scanning tunneling microscopy (UHVSTM) that InAs islands form in alignment along the monolayer step edges on (001) GaAs. Kitmura et al. (1995) demonstrated that InGaAs islands grown by MOCVD preferentially form on the bunched steps on the misoriented (001) GaAs substrates, and that selective island formation on the bunched step is possible. Growth on prepatterned substrates has been extensively studied as a way to control directly the in-plane alignment or position of islands. Preferential island formation has been found either on top of the ridges or along the sidewalls of the mesa stripes and at the bottom of the V-grooves, trenches, holes, and pits. Mui et al. (1995) demonstrated the self-alignment of InAs islands on etched GaAs ridges running along the [ O l l ] and [OlT] directions on the (100)-oriented substrates. InAs islands were formed on the sidewalls of ridges running along the [llO] direction, while for the ridges running along the [OlT] direction, islands were formed on the (100) plane on and at the foot of the mesa stripes. Moreover, as the grating pitch was reduced to 0.28 mm, islands were located either on the sidewalls or at the bottoms, with none on the tops. Seifert et al. (1996) reported InP islands on InGaP/GaAs overgrown stripes lithographically defined by metal stripes 30" off from the [TlO] direction on the (001) GaAs substrates. They demonstrated that the islands aligned either on top of the ridges, at the sidewall near the mesa edges, or at the bottom of the trenches, depending on the geometry of the InGaP/GaAs overgrown mesa stripes. Similarly results were obtained by Jeppensen et al. (1996) in an InP islands/InGaP/(OOl) GaAs system. They found that InAs islands grown by chemical-beam epitaxy form in chains with a minimum period of 33 nm along the trenches, and that single or a

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few islands that grow in an array in electron-beam lithographic holes on a (100) GaAs surface. Tsui et al. (1997) and Konkar et al. (1998) tried to form InAs islands selectively on top of the mesa stripes fabricated on (001) GaAs substrates. They demonstrated selective positioning of InAs islands on the top mesas depending on the geometry. The selective island formation on non-(001) prepatterned substrates were studied. Saitoh et al. (1996) used lithographically fabricated (wet-etched) tetrahedral-shaped recesses and V-grooves on (11 l)B GaAs substrates. They found that InAs islands selectively form at the bottom of those recesses. One of the most important purposes of the study of in-plane alignment of islands is improved structural uniformity. However, this purpose has yet to be achieved. We must further improve size and composition uniformity of islands grown on those stepped surfaces and grown by selective area growth. 111. Closely Stacked InAslGaAs Quantum Dots

The alignment of islands in the perpendicular direction will make it possible to couple islands electrically in the vertical direction as we reduce the intermediate GaAs thickness to the extent that the electron wave functions of neighboring wells are overlapped. This is the same situation as multiple quantum-well transfer to a superlattice when the barrier layer thickness is reduced. Vertical coupling enables electron tunneling between quantum dots, which lead to such novel electronic applications as a single-electron tunneling device. When we think of optical-device application of quantum dots, the most promising and practical advantage of the stacking technique will be size control. By choosing the intermediate-layer thickness and the repetition number of island layers, we will be able to tune the island size (height) and thus the quantized energies to meet device requirements. In addition, according to the discussion in Section 11, the size increase in the perpendicular direction will lead to a narrowing of the emission spectrum which will be quite beneficial for lasers. This section describes the growth process, the crystal structures, and the optical properties of perpendicularly stacked islands when the intermediate GaAs layer is reduced to a few nanometers close to or comparable to the height of InAs islands. Even under this condition, InAs islands are repeatedly grown. Optical diagnostics show that the stacking of the InAs islands increases the effective size of quantum dots in the perpendicular direction due t o electrical coupling, resulting in the narrowing of the spectrum FWHM to 25 meV. We call the stacked structure closely stacked quantum dots (Nakata et al., 1997).

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1. CLOSESTACKING OF InAs ISLANDS

Figure 2.10 shows the growth sequence for the close stacking of InAs islands. The amount of InAs supply for island formation was fixed at about 1.8 ML, and the nominal thickness of the GaAs intermediate layers was set at 2 and 3 nm. Prior to and following InAs island growth, the sample was annealed for 2 minutes and 1 minute, respectively. The growth rates, arsenic pressure, and growth temperature were the same as in Section 11.1. Figure 2.1 1 shows the RHEED-pattern intensity transition, observed at the area indicated by an arrow in the inset patterns, during the growth of InAs islands. The RHEED shows the streak pattern for the two-dimensional growth in the early stage, and the change to the spot pattern for the three-dimensional growth at above a critical amount of InAs supply. The SK-mode islands also grow for the 3rd and 5th layer. Note that the 3rd- and 5th-layer islanding started when the growth of InAs reached about 1 MLabout 63% of the 1st-layer islanding (approximated 1.6 ML). The reason for the smaller critical amount for the islanding is thought to be that the strain induced by the lower-layer islands accumulates InAs preferentially or that segregation of InAs from lower islands. The transition of the growth mode from two- to three-dimensional and the existence of wetting layers, as will be shown in the TEM image (Fig. 2.14(a)), both indicate that SK-growth islands were formed even on such thin GaAs intermediate layers.

FIG. 2.10. Growth sequence for the close stacking of InAs islands. The amount of InAs supply for island formation was fixed at about 1.8 ML, and the nominal thickness of the GaAs intermediate layers was set at 2 and 3nm. Prior to and following InAs island growth, the sample was annealed for 2 minutes and 1 minute, respectively.

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0

10

20

30

40

Time (s) FIG. 2.1 1. RHEED pattern intensity transition observed at the area indicated by an arrow in the inset patterns during the growth of InAs islands. The RHEED pattern shows that the SK island growth occurred also for the 3rd and 5th layer. Note that the 3rd- and 5th-layer island started when the growth of InAs reached about IOML, which was about 63% of the 1st-layer islanding.

The island size and the density at each layer were evaluated by ex-situ AFM. Figure 2.12 shows AFM images of the islanding surfaces at the (a) lst, (b) 3rd, (c) 5th, and (d) 10th layers stacked with 3-nm thick intermediate layers. These images refer to different epilayers grown under the same conditions. The scanned area is 250 x 250 nm'. The upper layer islands expanded slightly as the number of stacked layers increased. Figure 2.13 shows the dependence of island size on the number (diameter) and density of stacked layers. The average diameter of the 10th-layer islands was about 33 nm-40% larger than the 1st-layer islands, which were about 24 nm in diameter-while the island density decreased from 1 x loi1 cm-' for the 1st-layer islands to 8 x 10'ocm-2 for the 10th-layer islands. The increase in the diameter and decrease in the density were caused by the strain field formed by the lower islands as in Section 1.3. TEM photography shows the overall structural features. Figure 2.14(a) is a (110) cross-sectional TEM image of a 5-island stacked structure grown with 2-nm thick intermediate layers. Each island layer was accompanied by

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250 x 250 nm2 FIG. 2.12. AFM images of the islanding surfaces at the (a) Ist, (b) 3rd, (c) Sth, and (d) 10th layers stacked with 3-nm thick intermediate layers. The upper-layer islands expanded slightly as the number of stacked layers increased.

40

1.5

30

n

EC

U

10

0

0

2

6 8 Layer number

4

10

0 12

FIG. 2.13. The dependence of the island size (diameter) and density on the number of stacked layers. The average diameter of the 10th-layer islands was about 33 nm-40"/0 larger than the diameter of the 1st-layer islands (about 24 nm), while the island density decreased from 1 x 10" cm-' for the 1st-layer islands to 8 x IOL0cm-*for the 10th-layer islands.

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NAKATA AND

YOSHIHIRO SUGIYAMA

(a)

FIG.2.14. (a) ( I 10) cross-sectional TEM image of a 5-layer stacked island structure grown with 2-nm thick intermediate layers. Each island layer was accompanied by a wetting layer. The islands as a whole were about 22 nm in diameter and 13 nm in height, shown in the image as dark megaphone-like strained regions. (b) Plan view of a TEM image obtained from a 5-layer stacked island structure. Island density was 8 x 10'" cm-'. Size uniformity and lateral ordering were improved compared to a single-island layer.

a wetting layer, indicating that the upper-layer islands formed via SK growth, as did the 1st-layer islands. The upper-layer islands grew just on the lower-layer islands, aligning vertically. These islands as a whole had a structure of about 22 nm in diameter and 13 nm in height, shown in the image as dark megaphone-like strained regions. Figure 2.14(b) is a plan view of a TEM image obtained from a 5-layer stacked island structure grown with 2-nm thick intermediate layers. The density of the islands was 8 x 10'0cm-2, which agreed with the AFM results. What is surprising is that tht size uniformity and lateral ordering improved compared to the ordinary SK-mode islands (a single island layer). Although we attempted

2 MOLECULAR BEAMEPITAXIAL GROWTH

GaAs overgrowth

on lnAs islands

Annealing

(growth interruption)

137

2nd lnAs growth

FIG. 2.15. Growth process of stacked islands. The height of the InAs island after GaAs overgrowth decreased to the thickness of the intermediate layer. Then SK growth occurred on top of the lower island. This phenomenon enables us to precisely control the thickness of islands in the growth direction through the intermediate-layer thickness.

control only in the vertical dimension of the islands, uniform lateral dimension was also achieved in these closely stacked structures. Islands in each layer are seen to be spatially isolated in the vertical direction, with a 3- to 4-ML distance between the bottom of the upper-layer islands and the top of the lower-layer islands. The individual island was smaller in height than the intermediate GaAs layer thickness, and looked as if it were being buried in GaAs. The RHEED showed a streak pattern after overgrowth of GaAs and growth interruption, indicating that the interface between the overgrown GaAs and the upper InAs wetting layer is almost flat. A model to explain the formation process of this structure is illustrated in Fig. 2.15. First, the intermediate layers of GaAs overgrow away from the InAs islands, as confirmed by Xie et a]. (1995). During annealing (growth interruption after the GaAs overgrowth), the InAs of the upper part of the islands above the GaAs overlayers leave the islands and regrow to form parts of a wetting layer on GaAs. Then some of the InAs at the top regions in the remaining islands replaced with GaAs to reduce the total energy. This leads to a decrease in the island height. Similar features were also shown by Bimberg et al. (1996) and Ledentsov et al. (1996). This phenomenon enables us to precisely control the island height in the growth direction through the intermediate-layer thickness. The island-to-island distance of 3 to 4 M L is so thin that the electron wave functions of each island can be overlapped along the vertical direction. This suggests that the islands are electronically coupled and behave as a single quantum dot, which we will confirm by the optical diagnostics as follows.

2. PHOTOLUMINESCENCE PROPERTIES Light emission properties of these structures were evaluated using a photoluminescence technique with the same measurement conditions given

138

YOSHIAKI

0.9

NAKATA AND

YOSHIHIRO SUGIYAMA

I

I

I

I

I

1.0

1.1

1.2

1.3

1.4

Energy (eV)

0.9

1

1.1 1.2 Energy (eV)

1.3

1.4

FIG.2.16. (a) Emission spectra at 77 K of a single island layer and closely stacked 3-, 5-, and 10-layer island structures grown with 3-nm thick intermediate layers. The peak energy shifted to a lower energy as the number of stacked layers increased (about 90 meV in the 5-layer islands). At the same time, the broad emission spectrum of the single-island layer drastically narrowed at the 5-stacked layer to an FWHM of 27 meV-about one-third of that obtained for a single-layer island. (b) Interval layer thickness dependence on emission spectra with the five stacked layers. As the intermediate layer thickness decreased, the emission spectrum shifted toward lower energies and narrowed to an FWHM of 27 meV a t the 3-nm interval.

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in Section 111. Figure 2.16(a) shows the emission spectra at 77 K of a single island layer ( N = 1) and closely stacked 3-, 5-, and 10-layer island structures grown with 3-nm thick intermediate layers ( N = 3, 5, lo). The peak energy shifted to a lower energy as the number of stacked layers increased (about 90 meV in the 5-layer stacked islands). At the same time, the broad emission spectrum of the single-island layer drastically narrowed at the 5-layerstacked structure to an FWHM of 27 meV, which is about one-third of that obtained for a single-layer island. For a stack of 10 layers, the FWHM increased and integrated photoluminescence intensity decreased. This could be due to the increase in the total amount of strain around the stacked island structures, which induces dislocations and structural modulation. Figure 2.16(b) shows the dependence of the intermediate-layer thickness on emission spectra with the five stacked layers. As the intermediate layer thickness decreased, the emisson spectrum shifted toward lower energies and narrowed to an FWHM of 27 meV at the 3-nm thick intermediate layer. Figure 2.17 shows the excitation power dependence of the photoluminescence spectra for the 5-layer stacked islands with 2-nm (a) and 3-nm intermediate layers (b). As the excitation power increased, both samples exhibited peaks in the higher energy regime, which can be attributed to the higher-order quantized states. Note that energy separation between adjacent states was almost constant for both samples. The energy separation between the ground-state and the first excited-state emission was about 52 meV for the sample grown with 2-nm-thick intermediate layers and 44 meV for the sample grown with 3-nm-thick intermediate layers. The energy separation was smaller than that of the ordinary SK-mode islands (single-island layer) of about 70 meV, as seen in the electroluminescence spectrum (see Chapter 6). All the results observed above indicate that the electron wavefunctions between neighboring islands in the perpendicular direction overlap each other and that the stacked structure, as a whole, can work as a single quantum dot with a larger size. The proofs are (1) the red shift of the emission spectra with an increase in the repetition number; (2) the red shift of the emission spectra with a decrease in the intermediate-layer thickness; and (3) the decrease in the energy separation between the discrete energy states. As a result, we can control the quantum-dot size in the perpendicular direction by this close-stacking technique through the change in the stacking repetition number and in the intermediate layer thickness. The drastic improvement in spectrum broadening was just as expected. One direct reason for this success is that island size (height) is effectively increased, as seen above, decreasing the influence of dot size (height) fluctuation. Also, the island height is controlled more by the intermediatelayer thickness, as seen in Fig. 2.15. Finally, lateral dot size is more uniform, as seen in Fig. 2.14(b).

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0.9

1

1.1 1.2 Energy (eV)

1.3

1.4

1.3

1.4

(a)

0.9

1

1.1 1.2 Energy (eV)

(b) FIG.2.17. Excitation power dependence of photoluminescence spectra for the 5-layer islands with (a) 2-nm and (b) 3-nm thick intermediate layers. As the excitation power increased, both samples exhibited peaks in the higher energy regime, which can be attributed to the higher-order quantized states.

EXCITONCONFINEMENT EVALUATED BY 3. ZERO-DIMENSIONAL DIAMAGNETIC SHIFTS A question that often arises when we observe luminescence spectra from quantum-dot samples is whether the emission really originated from the dots and reflects the characteristics peculiar to their three-dimensional confinement. With closely stacked dots in particular, the identification is not so straightforward due to their structural complexity -namely, multiple stacked structure accompanied by wetting layers. The emission spectra in

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141

Section 111.2 indicated that the exciton states of the stacked dot form through the wave function overlap of each SK dot component, and that the perpendicularly stacked structure should be considered as one quantum dot. Magnetophotoluminescence, treated here, will give us not only the evidence of the three-dimensional confinement of excitons but also the exciton wave function extent in quantum dots. The principle can be summarized this way. The magnetic field confines excitons in the plane perpendicular to the field and increases their energy. We can determine the magnitude of other competing confinement potentials and thus the extent of the wave function by evaluating the number of energy shifts. The samples and the magnetic field direction are schematically shown in Fig. 2.18(a). The samples were five-layer-stacked InAs SK islands with 3-nm GaAs intermediate layers and, for comparison, an In,,,,Ga,~,,As/ GaAs quantum-well sample grown on a GaAs substrate by MOCVD. The In,,,,Ga,,,,As well-layer thickness was 8 nm and subjected to - 1.3% biaxial compressive strain. Figure 2.18(b) shows the photoluminescence spectra of the two samples with no magnetic field applied. Figure 2.18(c) shows the diamagnetic shifts for both samples under magnetic fields perpendicular and parallel to the sample plane at 4.2 K. In the quantum well, while the shift under a perpendicular field reached up to 7.3 meV at a maximum field of 11.8 T, the shift under the parallel field was much smaller (only 1.6meV at 11.8 T). The shift of the dots was almost independent of the field direction, and the maximum shift was 2.4meV in either direction. These results immediately give us the following insights into the difference in the quantum-confinement characteristics between the two samples. The asymmetrical shift in the quantum well is due to its onedirection quantum confinement. When the field is perpendicular to the plane, it works as a two-dimensional confinement potential for excitons in the well layer, causing large diamagnetic shifts. When the field is parallel to the plane, it affects only one direction in the quantum-well plane, since the other direction is already confined by strong potential barriers. The symmetrical, small diamagnetic shifts in the closely stacked dots show that excitons are three-dimensionally confined by an almost symmetrical confinement potential. The extent of the exciton wave function is estimated to be almost equal to the quantum-well thickness of 8 nm, since the shifts for the closely stacked dots and the quantum well under the parallel configuration are almost the same. Detailed quantitative analyses support this simple qualitative estimation (Sugawara et al., 1997). The extent of the wave function, approximately 8 nm, is smaller than the dot size observed by plan-view TEM (Fig. 2.14(a)) and definitely larger than the height of the single SK dot of about 3 nm. This indicates that the wave function of the ground-state exciton extends over

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Stacked islands (A = 1.088 (pm)) B

B

QW

(A = 0.936 (pm))

i;I

I

I

4.2 K

InGaAdGaAs QW

0.9

0.95

1.o

I

I

lnAslGaAs dot

*5.*meA

1.05

1.1

1.15

Wavelength (pm) FIG.2.18. (a) Samples for magneto-optical measurements. Five-layer stacked InAs SK islands with 3-nm GaAs intermediate layers and, for comparison, an In,~,,Ga,,,,As/GaAs quantum-well sample grown on a GaAs substrate by MOCVD. The In,,,,Ga,,,,As well layer was 8 nm thick and subjected to - 1.3% biaxial compressive strain. (b) Photoluminescence spectra of the two samples with no magnetic-field applied (from Sugawara et al., 1997. Copyright 1993 by The American Physical Society). (c) Diamagnetic shifts of emission spectra for both samples under magnetic fields perpendicular and parallel to the sample plane at 4.2 K (from Sugawara et al., 1997. Copyright 1993 by The American Physical Society).

2 MOLECULAR BEAMEPITAXIAL GROWTH

143

W W

0

20

40

60

A 80

100 120 140

Square of magnetic field ( T 2 ) FIG. 2.18. (c)

two or three SK dots out of five. Such exciton localization in the whole stacked structure can be attributed to the composition and/or size inhomogeneity of each SK dot components; that is, the potential minimum is formed by the coupling of the adjacent two or three SK dots with low band-edge energies.

IV. Columnar InAslGaAs Quantum Dots The advantages of the stacking technique for device applications can be summarized as follows. First, the spectrum width can be made much narrower than in ordinary SK dots. Second, the size (height) and the symmetry vary with the number of stacked layers, which makes it possible to artificially control the energy separation between the discrete quantum levels, the emission wavelength, the degeneracy of the quantum levels, the overlap integral of the electron-hole wave functions that determines the oscillator strength of optical transitions, and so forth. Though we succeeded in narrowing the spectrum width by close stacking of SK-mode islands and growth interruption (annealing) as explained in Section 111, we found that the emission efficiency was greatly damaged. The photoluminescence from the closely stacked islands rapidly weakened as the temperature rose and was barely observed at room temperature. Among

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various growth conditions, annealing after the growth of the GaAs intermediate layers (see the growth conditions in Fig. 2.10) was found to cause the most serious degradation. A large number of defects and/or impurities should have been introduced during annealing, working as nonradiative recombination centers. On the contrary, though the emission efficiency was fairly good even at room temperature without annealing, the spectrum width was reduced to 50-60 meV at most, presumably due to fluctuations in the island height and in the thickness of the spacer between the bottom of the upper-layer islands and the top of the lower-layer islands (Endoh et al., 1998). To overcome this problem, we stacked SK mode islands using even thinner intermediate layers -3 ML thick (less than 1 nm) -so that islands would physically contact each other in the perpendicular direction. This thickness is much less than the island height. As seen in Fig. 2.2, about 6 ML of GaAs supply was needed to fully cover the islands. The SK-mode islands could be grown even on such thin GaAs intermediate layers. Figure 2.19 shows the RHEED-pattern intensity transition during the stacked island growth (the GaAs intermediate layer on the first InAs island layers and the second InAs island layers). In this experiment, the first island layer was grown with the InAs supply of 1.8 ML. The intermediatelayer thickness was reduced to 3 ML from 10 ML, which is the same as that used in the closely stacked structures in Section 111. Immediately after the GaAs supply, the spot intensity decreased, showing that the growth surface was becoming flat by the growth of the intermediate layers. The spot intensity recovered to that of the first islands when the InAs supply started, showing that the second island layers grew. It should be noted that even on 3- and 5-ML thick intermediate layers, which were much thinner than the island height, second island layers grew as well as the first island layer. However, the critical amount of lnAs supply for the transition from two-dimensional layer growth to three-dimensional island growth decreased as the intermediate layer thickness has reduced: 0.84ML for the 10-ML intermediate layers, 0.76 ML for the 7-ML intermediate layers, 0.60 ML for the 5-ML intermediate layers, and 0.26ML for the 3-ML intermediate layers. These were much smaller than that for the first island layer, shown at the bottom of Fig. 2.19. Figure 2.20 shows the RHEED-pattern intensity transition when the InAs islands were repeatedly stacked using the 3-ML thick intermediate layers. The inset photographs are the AFM images of the topmost layer after 8 to 9 repetitions. The amount of InAs supply for the stacked-island formation was changed from about 0.5 ML to 0.8 ML. When the stacked islands were grown with an InAs supply of about 0.7ML, the RHEED intensity oscillated with the InAs island growth (RHEED intensity increased) and

2 MOLECULAR BEAMEPITAXIAL GROWTH

Growth time (s)

145

20 s I 1 div.

FIG.2.19. RHEED pattern intensity transition during the stacked island growth (GaAs intermediate layer o n the first InAs islands and the second InAs island). The critical thickness for the transition from two-dimensional growth t o island growth decreases as the intermediate GaAs layer thickness reduces.

GaAs intermediate-layer growth (RHEED intensity decreased). This suggests that the islands were formed successfully. The AFM image of the topmost island layer grown with the 0.7-ML InAs supply shows that stacked islands were formed with the same size and density as in the first islands. On the other hand, when stacked with an InAs supply of about 0.5 ML and 0.8 ML, RHEED intensity oscillation stopped. When stacked with the InAs supply of about 0.5 ML, RHEED intensity damped at the third repetition, and the AFM image showed no island formation. This is possibly because the InAs supply was not enough for island formation. When stacked with an InAs supply of about 0.8 ML, RHEED intensity did not recover completely at 8 to 9 repetitions and islands like the first one were not seen in the AFM image. The supply of 0.8 ML is excessive. The optimum amount of about 0.7 ML is smaller than in the case of ordinary SK-mode islands. When stacked islands using such thin intermediate layers, the amount of InAs supply has to be reduced and optimized precisely.

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a

YI U

Growth time ( 5 )

40 8 I 1 div.

FIG.2.20. RHEED pattern intensity transition when the InAs islands were repeatedly grown with the intermediate layer of 3 ML. The inset photograph shows the AFM images of the topmost layer. The above results suggest that there is an optimum lnAs supply amount.

Figure 2.21(a) is a cross-sectional TEM image of the structure shown schematically in Fig. 2.21(b). InAs island layers formed with an InAs supply of 0.7 M L and GaAs intermediate layers of 3 M L were grown alternately (8 repetitions) on the first island layer formed with an 1.8-ML InAs supply. The stacked upper islands were grown on the lower-layer islands aligning perpendicularly on the first-layer islands. The islands on each layer were in contact with each other physically, and the stacked islands as a whole had a columnar shape with a diameter and a height of about 17 nm and 13 nm, respectively. All island layers were accompanied by wetting layers, seen as horizontal line-shaped contrasts in the TEM image, indicating that the stacked islands were formed by the SK mode. Figure 2.21(a) is a low-magnification image. The columnar-shaped islands were seen clearly in the quantum-well-like dark contrast region composed of lnAs multiple wetting layers and GaAs intermediate layers. In Chapter 3, we will introduce InGaAs/GaAs quantum dots with a light emission wavelength of 1.3 pm grown by the alternate growth of elementary In-Ga-As in MOCVD. The present structures look very similar to the ALS dots. Although the large islands were formed, the contrasts corresponding to dislocations and stacking faults, which observed at the coalesced large islands, as shown in Fig. 2.6, were barely observed, suggesting that the composition of lnAs was smaller than that in the ordinary SK islands.

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147

FIG.2.21. (a) Cross-sectional TEM image of columnar-shaped quantum dots. (b) Schematic structure. (c) Low magnification image.

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YOSHIAKI

NAKATA AND

YOSHIHIRO SUGIYAMA

Figure 2.22 shows the photoluminescence spectrum of these columnarshaped quantum dots, measured at 300K. The peak wavelength of the ground level was 1.17pm, and that of the excited level was 1.10pm. An emission peak related to the multiple wetting layers was observed at 1.01 pm. An FWHM of the spectrum was about 42meV. This is much smaller than that for the ordinary SK-mode quantum dots of about 80 meV, indicating that structural uniformity was fairly improved. The photoluminescence intensity was also better than the SK-mode quantum dots and was over 1000 times stronger than in the closely stacked quantum dots described above, suggesting that the introduction of defects and impurities working as the nonradiative recombination centers was remarkably suppressed. The emission from the wetting layers red-shifted compared with the ordinary SK mode quantum dots. This is because multiple wetting layers were coupled electrically and formed a quantum level as a whole. The energy separation between the ground state and the wetting layer was 168meV, which is much smaller than 230meV of the ordinary SK-mode quantum dots. An excellent performance of columnar-shaped quantum dots is demonstrated in Chapter 6. Figure 2.23 plots the photoluminescence wavelength at room temperature and 7 7 K as a function of the stacked-layer number. As the layer number

0.9

1.o

1.1 1.2 Wavelength (pm)

1.3

1.4

FIG.2.22. Photoluminescence spectrum of the columnar-shaped quantum dots shown in Fig. 2.21 measured at 300 K. The peak wavelength of the ground level was 1.17 pm and that of the excited level was 1.10 pm. An emission peak related to the multiple wetting layers was observed at 1.01 p m A full width at half maximum (FWHM) of the spectrum was about 42 meV.

2 MOLECULAR BEAMEPITAXIAL GROWTH

t

0.9’ 0

149

i I

10

’

20

I

I

30

Stacked-layer number

”

40

FIG.2.23. Photoluminescence wavelength at room temperature and 77 K as a function of the stacked layer number. As the layer number increases, the emission wavelength shifts to longer wavelength due to the size increase.

increased, the emission wavelength becomes longer due to the size increase. It is surprising that the longest wavelength at room temperature was 1.24 ym at the 23rd stacking- very close to the practically applicable 1.3 pm. Structural features of the stacked islands are summarized in Fig. 2.24. Stacked-island structures strongly depend on the intermediate layer thickness, t (nm). When t > 20 nm, stacked islands form without any correlation to the lower-layer islands. The mean island size and density are constant in all island layers, which is useful for increasing island density per unit volume. When t < 20 nm, stacked upper islands form in correlation to the lower-layer islands due to strain fields induced by the lower-layer islands. The upper-layer islands form in alignment with the first-layer islands in the perpendicular direction, and expand with an increase of the stacked-layer number. As the intermediate-layer thickness decreases almost to the island height, the size expansion of the stacked islands is suppressed, and almost equal size islands are stacked closely. In this case, the electron wave functions between neighboring islands in the perpendicular direction overlap each other, and the perpendicularIy stacked structure works effectively as a single large quantum dot. Drastic improvement of the spectrum linewidth is observed. When the intermediate-layer thickness further de-

150

0

YOSHIAKI NAKATAAND YOSHIHIROSUGIYAMA

10 20 30 Intermediatelayer thickness, t (nm)

FIG. 2.24. Structural features of stacked islands. Stacked island structures strongly depend on the intermediate-layer thickness, t (nm). When f z 20nm, stacked islands were formed without any correlation to the lower-layer islands. When t < 20 nm, stacked upper islands were formed in correlation to the lower-layer islands due to strain field induced by the lower-layer islands.

creases to a few monolayers of less than 1 nm, which is much smaller than the island height, stacked islands are formed. This success is achieved by optimizing the amount of InAs supply for the stacked-island formation. If we use 3-ML-thick intermediate layers, the amount of InAs supply has to be reduced to about 0.7ML. The islands in the stacked structure are in contact with each other physically in the perpendicular direction and the stacked structure has a columnar shape as a whole. Even when we use these thin intermediate layers, all stacked islands are accompanied by wetting layers, indicating that the islands grow in the SK mode.

V. Summary The structural features and optical properties of quantum dots presented in this chapter are summarized in Fig. 2.25. Ordinary InAs SK-mode quantum dots on GaAs substrates show broad spectra with a typical FWHM of 80meV because of their large structural fluctuation. Since the shape of ordinary SK-mode quantum dots is rather flat, their spectrum broadening depends mainly on the fluctuation of the island height. The stacking techniques to grow the closely stacked and columnar-shaped dots

2

151

MOLECULAR BEAMEPITAXIAL GROWTH SK islands

Close1 stacked isinds

Columnar-shaped islands

90 meV

25 meV

40 meV

Good

Poor

Good

Schematic structure PL FWHM

PL efficiency In-plane coverage (Density) Wavelength (300K)

40%

40%

> 1 x 10'1 cm-2

> 1 x 1011 cm-*

Clm

1.2 pm

40%

>1 x

loll

cm-2

> 1.2 pm

Frc. 2.25. Structural and optical properties of different types of dots

are very useful in suppressing island height fluctuation because the effective height can be controlled artificially by the number of stacked island layers. The luminescence spectrum was remarkably improved from 80 meV to 25 meV by the close stacking method. In columnar-shaped quantum dots, where both narrow spectrum width and high emission efficiency are obtained, we have successfully achieved low-threshold and highly efficient operation of quantum-dot lasers (Chapter 6). According to the simulation of quantum-dot laser performance in Chapter 1, the narrow spectrum width we have achieved is very promising for high-performance quantum-dot lasers that are superior to strained quantum-well lasers. If we aim at high-speed direct-modulation quantum-dot lasers, as predicted by Arakawa and Sakaki (1982), we need to further reduce the spectrum width to 10 meV (Chapter 1). The in-plane size fluctuation of our columnar-shaped dots was evaluated by AFM and found to give a broadening of about 20meV (Endoh et al., 1998). For that reason, we are now concentrating on improving in-plane size homogeneity.

Acknowledgments The authors would like to thank Dr. Hiroshi Ishikawa, Dr. Hajime Shoji, Dr. Kohki Mukai, Dr. Osamu Ueda, Dr. Akira Endoh, and Dr. Toshiro Futatsugi, of Fujitsu Laboratories Ltd., and Professor Shunichi Muto of

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Hokkaido University for their fruitful input and support. We also are grateful to Dr. Hajime Ishikawa and Dr. Naoki Yokoyama of Fujitsu Laboratories Ltd., for their encouragement and strong support throughout this work.

REFERENCES Arakawa, Y., and Sakaki, H. (1982). Appl. Phys. Lert. 40, 939. Arita, M., Avramescu, A., Uesugi, K., Suemune, I., Numai, T., Machida, H., and Shimoyama, N. (1997). Jpn. J . Appl. Phys. 36. 4097. Bimberg, D., Ledentsov, N. N., Grundmann, M., Kirstaedter, N., Schmidt, 0.G., Mao, M. H., Ustinov, V. M., Egorov, A. Y., Zhukov, A. E., Kop’ev, P. S., Alferov, Z . I., Ruvimov, S. S., Gosele, U., and Heydenreich, J. (1996). J p n . J. Appl. Phys. 35, 1311. Carlsson, N., Georgsson, K., Montelius. L.. Sarnuelson, L., Seifert, W., and Wallenberg, R. (1995). J . Cryst. Growth 156, 23. Endoh, A., Nakata, Y., Suguyama, Y., Takatsu, M., and Yokoyama, N. (1998). Pmc. 10th Inr. Conf: InP and Related Materials. Fafard, S., Leon, R., Leonard, D., Martz, J. L., and Petroff, P. M. (1994). Phys. Reu. B. 50,8086. Fukui, T., and Saito, H. (1987). Appl. Phys. L m . 50, 824. Gershoni, D., Ternkin, H., Dolan, G. J., Dunsmuir, J., Chu, S. N. G., and Panish, M. B. (1998). Appl. Phys. Lett. 53, 995. Goni, A. R., Pfeiffer, L. N., West, K. W.. Pinzuk, A., Baranger. H. U., and Storrner, H. L. (1992). Appl. Phys. Lett. 61, 1956. Grundrnann, M., Christen, J., Ledentsov. N. N., Bohrer, J., Bimberg, D., Ruvirnov, S. S., Werner, P., Richter, V., Gosele, V., Heydenreich, J., Ustinov, V. M., Egorov, A. Y., Zhukov, A. E., Kopev, P. S., and Alferov, 2 . 1. (1995a). Phys. Rev. Lett. 20, 464. Grundmann, N.. Ledentsov, N., Heitz, R.. Eckey, L., Christen, J., Bohrer, J., Bimberg, D., Ruvimov, S. S., Werner, P., Richter, V., Heydenreich, J., Ustinov, V. M., Egorov, A. Y . , Zhukov, A. E., Kopev, P. S., and Alferov, Z. 1. (1995b). Phys. Stat. Sol. 188, 249. Glas, F., Guill, C., Henoc, P., and Haussy, F. (1987). Inst. Phys. Can$ Ser., No. 87; Section 2, p. 87. Grundmann, M., Ledentsov. N. N., Steir, 0..Bimberg, D., Ustinov, V. M. Kop’ev, P. S., and Alferov, Z. I. (1996). Appl. Phys. Lert. 68. 979. Hessman, D.. Castrillo, P., Pistol, M.-E.. Pryor. C., and Samuelsson, L. (1996). Appl. Phys. Lett. 69, 749. Houzay, F., Guille, C., Moisson, J.-M., Henoc, P., and Barth, F. (1987). J . Cryst. Growth 81,67. Ikoma, N., and Ohkouchi, S. (1995). J p n . J . Appl. Phys. 34, L724. Jeppensen, S.. Miller, M. S., Hessrnan. D., Kowalski, B., Maximov, I., and Sarnuelson, L. (1996). Appl. Phys. Lett. 68, 2228. Kapon, E.. Kash, K., Olausen, E. M.. Hwang, D. M., and Colas, E. (1992). Appl. Phys. Lett. 60, 477. Kapon, E., Simhony, S., Bhat, R., and Hwang. D. M. (1989). Appl. Phys. Lett. 55, 2715. Kitamura, M., Nishioka, M., Oshinowo, J., and Arakawa, Y. (1995). A p p l . Phys. Lett. 66, 3663. Konkar, A. Madhukar, A,, and Chen, P. (1998). Appl. Phys. Lett. 72, 220. Ledentsov, N. N., Bohrer, J., Bimberg. D., Kochnev, 1. V., Maximov, M. V., Kop’ev, P. S., Alferov, 2 . I., Kosogov, A. O., Ruvimov. S. S., Werner, P., and Gosele, U. (1996). Appl. Phys. Lett. 69, 1095.

2 MOLECULAR BEAMEPITAXIAL GROWTH

153

Leon, R., Petroff, P. M., Leonard, D., and Fafard, S. (1995). Science 267, 1966. Leonard, D., Krishnamurthy, M., Reaves, C. M., Denbaars, S. P., and Petroff, P. M. (1993). Appl. Phys. Lett. 63, 3203. Leonard, D., Pong, K., and Petroff, P. M. (1994). Phys. Rev. B. 50, 11687. Lewis, B. F., Lee, T. C., Grunthaner. F. J.. Madhukar, A,, Fernandez, R., and Masanian, F. F. (1984). J . Vac. Sci. Techno/. B2, 419. Marzin, J.-Y., Gerard, J.-M., Izrael, A,, and Barrier, D. (1994). Phys. Rec. Lett. 73, 716. Marzin, J.-Y., Gerard. J.-M., Israel, A., Barrier, D., and Bastard, G. (1994). Phys. Rer. Lett. 73, 716. Miyamoto, Y., Cao, M., Shingai, Y., Furuya, K., Suematsu, Y., Ravikumar, K. G., and Arai. S. (1987). Jpn. J . Appl. Phys. 26, L225. Moison, J. M., Houzay, F., Barthe, F., Leprince, L.. Andre, E., and Vatel, 0. (1994). A p p l . Phys. Lett. 64, 196. Mui, D. S., Leonard, D., Coldren. L. A,. and Petroff, P. M. (1995). Appl. Phys. Lett. 66. 1620. Mukai, K., Ohtsuka, N., Sugawara, M., and Yamazaki, S. (1994). J p n . J . Appl. Phys. 33. L1710. Muto, S. (1995). Jpn. J . Appl. Ptiyr. 34. L210. Nakata, Y.. Sugiyama, Y., Futatsugi. T,. and Yokoyama, N. (1997). J . Cryst. Growth 175/176, 173. Nishi, K., Anan, T., Gomyo, A,. Kohmoto, S.,and Sugou, S. (1997). Appl. Phys. Lett. 70. 3579. Oshinowo, J., Nishioka, M., Ishida, S.. and Arakawa, Y. (1994). Appl. Phys. Lett. 65, 1421. Petroff, P. M., Gossard, A. C., Logan, R. A., and Wiedmann, W. (1982). Appl. Phys. Lett. 41, 635. Petroff, P. M., Gossard, A. C., and Wiegmann, W. (1984). Appl. Phys. Lett. 45. 620. Pfeiffer, L. N., Stormer, H. L., Baldwin. K. W., West, K. W., Goni, A. R., Pinczuk, A. Ashoori, R. C., Dignarn, M. M., and Wegscheider, W. (1993). J . Cryst. Growrh 127. 849. Reaves, C. M., Bressler-Hill, V., Varma, S.. Weinberg, W. H., and DenBaars, S. P. (1995). Surf Sci. 326. 209. Reed, M. A., Bate, R. T., Bradshow. K.. Duncan, W. M., Frenseley, W. R., Lee, J. W., and Shih, H. D. (1986). J . Vuc. Sci. Echnol. B4. 358. Saitoh, T., Taniniura, A., and Yoh. K. (1996). J p n . J . Appl. Phys. 35, 1370. Schaffer. W. J., Lind, M. D., Kowalczyk, S. P., and Grant, R. W. (1983). 1. Vac. Sci. Trchnot. B1, 688. Seifert, W., Carlsson, N., Peterson, A.. Wernersson, L.-E., and Samuelson, L. (1996). Appl. Phjs. Lett. 68, 1684. Shchukin, V. A,, Ledentsov, N. N., Kop’ev, P. S., and Bimberg, D. (1995). Appl. Phys. Lett. 75, 2968. Solomon, G. S., Komarov, S. K.. Harris. J. S., Yamamoto, Y. (1997). J . Cryst. G r o w h 175/176. 707. Solomon, G. S., Trezza, J. A., Marshall, A. F., and Harris, J. S. (1996). Pkjs. Reu. Lett. 76. 952. Stranski, 1. N., and Krastanow, L. (1937). Aktid. Wiss. Wien Math.-Natur. ilh 146. 797. Tackeuchi, A., Nakata, Y.. Muto, S., Sugiyama, Y., Inata, T., and Yokoyama. N. (1995). J p n . J . Appl. Phys. 34, L405. Tabuchi, M., Noda, S., and Sasaki, A. (1992). Sci. & Tech. Mesoscopic Structures 379. Tanaka, S.. Hirayama, H., Ramvall, P.. and Aoyagi, Y. (1997). Piocec~cli,zgs of the 2nd Inrernationul Corzference on Nitride Seniiconductors, 48. Tanaka, S., Iwai, S., and Aoyagi, Y . (1996). Appl. Plzys. Lett. 69, 4096. Tersoff, J., Tieichert, C., and Lagally, M. G. (1996). Phys. Rev. Lett. 76, 1675. Tsui, R., Zhang, R., Shiralagi, K., and Golonkin. H. (1997). Appl. Phys. Lett. 71, 3254. Tu,, K.-N., Mayer, 3. W., and Feldnian. C. (1992). “Electronic Thin Film Science,” Chapter 7. Macmillan Publishing Company.

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Ueda, O., Nakata, Y., and Muto, S. (1998). Prcir. 10th Int. ConJ InP and Related Muterial.?, p. 113. Wegscheider, W., Pfeiffer, L. N., Pinczuk. A., West, K. W., Dignam, M. M., Hull, R., and Leibenguth, R. E. (1995). f.Cryst. Growth 150, 285. Xie, Q., Madhukar, A,, Chen, P., and Kobayashi, N. P. (1995). Phys. Rev. Lett. 75, 2542.

SEMICONDUCTORS AND StMIMETALS. VOL 60

CHAPTER 3

Metalorganic Vapor Phase Epitaxial Growth of Self-Assembled InGaAs/GaAs Quantum Dots Emitting at 1.3 ,urn Kohki Mukai, Mitsuru Sugawara, and Mitsuru Egawa OPTICAL SEMICONVVCTOR

D E V I C H LAB<>RATORY

FUJITSU LABORATORIFS LTD. ATSUGI, KANAGAWA. JAPAN

Nobuyuki Ohtsuku [NTGRATtU

MATERIALS LABVRATORY

FUJITSULABORATORIES LTU.

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1.

VTRODUCTION

. . . . . . . . . . . . . . . . . . . . . . . . . .

I

11. ATOMICLAYEREPITAXIAL GROWTH. . . . . . . . . . . . . . . . . . . 111. ALTERNATE SUPPLY GROWTH OF InGaAs DOTS BY In-As-Ga-As SEQUENCE . . IV. ALTERNATE SUPPLY GROWTH OF InGaAs DOTSBY THE In-Ga-As SEQUENCE . . 1. Tiiw Types of ALS Do/ . . . . . . . . . . . . . . . . . . . . . . .

2. Multiple-Loyer Growfh . . . . . . . . . . . . . . . . . . . . . V . THEGROWTH PROCESS. . . . . . . . . . . . . . . . . . . . . . VI. S U M M A R Y. . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

,

,

. .

. . . . .

157 160 166 168 172 176 180 181

I. Introduction The molecular beam epitaxial (MBE) growth of InAs self-assembled quantum dots was described in Chapter 2. This growth process occurs in the well-known Stranski-Krastanov (SK) mode, which produces threedimensional islands accompanied by a thin wetting layer to release the strain energy accumulated by the large lattice mismatch between the GaAs substrate and the InAs. Typical InAs self-assembled islands have a flat shape with a base length of about 20nm and a height of a few nm. Since the exciton Bohr radius is 10 to 20nm in the InAs-GaAs system, the island is 155 Copyright i :1999 by Academic Press All rights of reproductioii in any form reserved. ISBN 0-12-752169-0 ISSN 0080-8784 99 $3000

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KOHKIMUKAI, MITSURU SUGAWARA, AND MITSURU EGAWA

small enough to observe the three-dimensional quantum confinement effect; that is, the islands work as quantum dots. The perpendicularly stacked InAs islands, named closely stacked quantum dots and columnar quantum dots in Chapter 2, have enabled us not only to tune the emission wavelength by increasing the dot size primarily in the perpendicular direction, but also to narrow the spectrum linewidth by reducing the fluctuation of island size. In particular, columnar quantum dots have had great success in achieving low-threshold and highly efficient quantum-dot lasers, as described in Chapter 6 and late news, due to their high crystal quality. A new type of self-assembled quantum dots, grown by metalorganic vapor phase epitaxy (MOVPE), is introduced in this chapter (Mukai et al., 1994; Ohtsuka et al., 1995). Their growth sequence is unique in that the group-111 and group-V precursors are supplied alternately with an amount corresponding to one or less than one monolayer. One of the most striking features of our original quantum dots is that they emit at the wavelength of 1.3 pm at room temperature, which is the zero-dispersive wavelength of the silica optical fiber used in optical data transmission systems. Considering that the emission wavelength of ordinary SK dots is typically 1.1-1.2pm, the longer emission wavelength suggests larger incorporation of In atoms into quantum dots and/or larger dot size. In addition, the emission spectrum linewidth is very narrow and gives a series of distinct lines peculiar to three-dimensional quantum confinement, as seen in Fig. 4.1 l(a) in Chapter 4. The minimum full width at half maximum (FWHM) is 28meV-much smaller than the 80-120meV in the SK-growth islands. These two features- 1.3-pm emission and narrow spectrum linewidth -make quantum dots very attractive for practical applications. If we succeed in 1.3-pm lasing from quantum dots, strained quantum-well lasers -currently the standard device in optical transmission systems -might be replaced by high-performance quantum-dot lasers. Vertical-cavity surface-emitting lasers (VCSELs) with 1.3-pm quantum-dot active regions are also attractive applications, since high-reflectivity distributed Bragg reflector mirrors can be fabricated on GaAs substrates. We named our original materials ALS quuntum dots, for Alternate Supply. We found these new quantum dots when we tried to fabricate (GaAs),/ (InAs), short-period superlattices on GaAs substrates by atomic layer epitaxy (ALE). The purpose of our research was to realize materials that emit at 1.3pm on GaAs substrates using the ALE technique, with the expectation that lasers on GaAs substrates would have high temperature stability since high-potential barriers like AlGaAs and AlGaInP can prevent carrier leakage from the active region. Simply growing InGaAs quantum wells on GaAs substrates does not work because the large lattice mismatch causes misfit dislocations, severely damaging the crystal quality. Our belief

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was that short-period superlattices can break through the limit set by the critical thickness to reach a highly efficient 1.3-pm emission. Actually, Roan and Cheng (1991) had already succeeded in growing (GaAs),/(InAs), short-period superlattices on GaAs substrates by MBE that showed 1.3-pm emission at room temperature. Using the ALE approach, we also succeeded in realizing 1.3-pm emission and in making the emission spectrum linewidth much narrower (Ohtsuka and Ueda, 1994). However, we found that the grown materials were not short-period superlattices but had bizarre structures, as seen in the transmission electron microscope (TEM) photograph of Fig. l.l(b) in Chapter 1, where spherical dark regions are buried in the quantum wells. Having done various diagnostics of the materials (see Chapter 4), we finally concluded that the structures are quantum dots. This chapter reviews ALS quantum-dot growth. We start by describing the self-limiting one-monolayer growth of InAs and GaAs and the growth of (InAs),/(GaAs), superlattices. Then we present two kinds of growth sequences for the quantum dots: one, an In-As-Ga-As sequence based on the concept of making monolayer superlattices by ALE; the other, a newly developed In-Ga-As sequence. Quantum dots are characterized by their shape, size, composition, and emission wavelength. In particular, we illustrate how the dots vary with alternate supply cycle, growth temperature, and composition of the buffer layers on which the dots are grown. Finally, we compare the growth process of ALS quantum dots with that of SK dots.

11. Atomic Layer Epitaxial Growth

As stated in the introduction, our original purpose was to grow shortperiod superlattices using the ALE technique. Because we found that the grown materials were quantum dots, we tried to refine various growth conditions to improve crystal quality. As we pursued dots with higher emission efficiency, we changed the growth sequence from In-As-Ga-As, according to the concept of superlattice atomic layer epitaxy, to In-Ga-As, and we reduced the supply amount of sources to less than one monolayer. However, the alternate supply similar to ALE growth is definitely a key to producing this type of quantum dot. Another inheritance from ALE growth is trimethylindium-dimethylethylamineadduct (TMIDMEA), which we developed as a source material for indium. Here, we will briefly describe the self-limiting growth of GaAs, InAs, and short-period superlattices. For ALE growth, we used pulse jet epitaxy (PJE) with a chimney reactor, in which precursors are supplied in a fast pulsed stream, and self-limited 111-V compounds can be grown under a wide range of conditions (Ozeki et

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al., 1988, 1989, 1991; Sakuma et al., 1988). The structure of the growth system is shown in Fig. 3.1. The reactor was designed through a simulation to realize smooth gas flow streamlines without vortices. We adopted a low-pressure atmosphere for precursors to decompose only on the heated substrate surface. The substrate was positioned in the fast gas stream emitted from a jet nozzle to prevent thermally decomposed source molecules from building up in stagnant layers. Source materials were trimethylindium (TMIn), TMIDMEA, trimethylgallium (TMGa), triethylgallium (TEGa), and arsine (ASH,) diluted with hydrogen (HJ. Experiments on self-limiting ALE growth were done on GaAs substrates for GaAs growth and on InAs substrates for InAs growth. The most severe problem we encountered in fabricating InAs/GaAs superlattices using ALE was that the temperature range for self-limiting growth was different for the two binary compounds-450 to 550°C for GaAs using TMGa and 300 to 400’C for InAs using TMIn. Since there is no overlap in ALE growth temperatures for the two materials, it appeared difficult to fabricate the superlattices under precise self-limiting growth (Mori and Usui, 1992). A key factor in determining the temperature range for self-limiting growth is the decomposition temperature of the group-I11 precursors. We expected that the thermal stability of TMIn, which is a

Substrate [InAs (IOO)]

FIG. 3.1. Growth system for pulse-jet epitaxy with a chimney reactor, in which precursors are supplied in a fast, pulsed stream and self-limited 111-V compounds can be grown under a wide range of conditions.

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conventional In source, might be sustained at higher temperatures by the addition of some adduct, and so developed TMIDMEA as a new In source for ALE. The ALE growth of InAs using TMJDMEA and TMIn for indium sources is shown in Fig. 3.2, where the relationship between the group-I11 pulse duration and the growth thickness per cycle is plotted at 460°C. When we used TMIn, the thickness per cycle increased monotonously with the TMIn pulse duration, because TMIn decomposes in the boundary layer at this high temperature and a self-limiting mechanism does not work. When we used TMIDMEA, complete self-limiting growth could be achieved -the growth thickness per cycle saturated at one monolayer of InAs. The temperature dependence of the growth rate is shown in Fig. 3.3. The self-limiting growth of InAs using TMIDMEA was observed over a temperature range of 350 to 500°C. The upper limit of this temperature region is 100°C higher than that using TMIn. As a result, we can grow InAs and GaAs with a self-limiting mechanism in the common temperature region, which makes it possible to grow InAs/GaAs heterostructures at a constant growth temperature.

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1

300

400

500

600

Growth temperature (“C) FIG.3.3. InAs and GaAs thickness per cycle of ALE growth as a function of growth temperature. We can grow InAs and GaAs with a self-limiting mechanism in the common temperature region. This makes it possible to grow InAs/GaAs heterostructures at a constant growth temperature (from Ohtsuka and Ueda. 1994).

Using this technique, we grew (InAs),/(GaAs), superlattices on a GaAs substrates. The grown sample consisted of a 500-nm-thick undoped GaAs buffer layer, a 12-period (InAs),/(GaAs), layer with a total thickness of 7nm, and a 30-nm thick undoped GaAs cap layer. Both buffer and cap layers were grown by a conventional MOVPE at 460°C using TEGa. The room-temperature photoluminescence spectrum of the sample grown using TMIDMEA showed a peak at 1.34pm with an FWHM of 28 meV (Fig. 3.4). The emission intensity was much stronger than the sample grown using TMIn. The transmission microscope (TEM) photograph of the “superlattice” surprised us. We discovered quantum dots in the alternately supplied layer, as shown below.

111. Alternate Supply Growth of InGaAs Dots by In-As-Ga-As Sequence In this section, we illustrate the growth of ALS dots by In-As-Ga-As alternate supply using the same reactor for PJE and the same sources as

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used in Section 11. Each supply is separated by H, purging pulses. The dots were grown on an In,Ga, -,As buffer layer on a (001)-GaAs substrate and were covered by an In,Ga, -,As layer having the same composition as the buffer layer. The supply amount of TMIDMEA and TMGa was set at less than one monolayer to effect higher emission efficiency. Figure 3 4 a ) and (b) show typical plan-view and cross-sectional TEM images of ALS dots, grown by 12 cycles of (InAs),/(GaAs), periodic supply on a 300-nm GaAs buffer layer on a (001)-GaAs substrate, covered by a 30-nm-thick GaAs cap layer. The growth temperature was 460°C which provides distinct self-limiting growth for both InAs and GaAs (see Fig. 3.3). The growth pressure was 15 Torr. A plan-view image shows uniform dot-like microstructures about 20 nm in diameter and an area coverage of 5-10%. The dots diameters follow a Gaussian distribution with a standard deviation of 2.9 nrn. A cross-sectional dark-field image indicates that sphereshaped dots were self-assembled within the InAs/GaAs short-period layer (dot layer) sandwiched by GaAs. They were about 10nm in height and surrounded laterally by a quantum-well layer having almost the same thickness as that of the dots. The thickness of the quantum-well layer was definitely greater than that of the so-called wetting layer of the SK islands (Krost et al., 1996; Ramachandran et al., 1997a, and b. Kitamura et al.,

162

KOHKIMUKAI.MITSURUSUGAWARA, AND MITSURU EGAWA

FIG.3.5. (a) Plan-view and (b) cross-sectional TEM images of ALS dots, grown by 12 cycles of (InAs),/(GaAs), periodic supply on a 250-nm GaAs buffer layer on a (001) GaAs substrate and covered by a 30-nm-thick GaAs cap layer.

1997). We observed two clear upper and lower borders between the GaAs sandwiching layers and the dot layer. We also observed a high-resolution TEM image around the dot in a cross-sectional sample and did not find any dislocations or defects in the lattice image (Fig. 3.6). A weak contrast at the border of the dots suggests that the composition and lattice distortion do not abruptly change around the dot. Figure 3.7 shows the spatial composition distribution of a sample cross section evaluated by energy-dispersive X-ray microanalysis (EDX) with an excitation electron beam focused to a diameter of about 1 nm. The sample was thinned down to lOnm, which is less than the in-plane diameter of the dots, to obtain a signal only from the dots. Signal intensity was calibrated to estimate a precise indium composition using the signal intensity of a lattice-matched In,,,,Ga,~,,As layer grown on an InP substrate as a reference. The indium composition was 0.5 at the center of the dots (i.e.,

163

3 METALORGANIC VAPORPHASEEPITAXIAL GROWTH

FIG. 3.6. High-resolution T E M image around the dot in a cross-sectional sample. No dislocation or defects are found in the lattice image.

Ino~,Gao,,As)and 0.1 in the quantum-well layer surrounding the dots (i.e., Ino.,Gao.!Js). Our self-assembled quantum dots were found to have a unique structure, different from that of SK islands. For one thing, ALS dots are almost buried in the quantum-well layer with flat upper and lower interfaces, and they are not accompanied by a so-called wetting layer. For another, the proportion of height and diameter is different from that of the SK island; that is, the height of the SK dots is generally less than one-fourth of the in-plane

0.27

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Indium composition FIG.3.7. Spatial indium composition distribution of quantum-dot cross sections evaluated by energy dispersive X-ray microanalysis (EDX) with an excitation electron beam focused to a diameter of about 1 nm (from Mukai et al., 1994).

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KOHKIMUKAI, MITSURU SUGAWARA, AND MITSURU EGAWA

diameter, while the height of ALS dots is about half of the in-plane diameter. These characteristics suggest that our dots were not self-assembled by the conventional SK mode. The outstanding characteristic of our quantum dots is a wide-range wavelength tunability of between 1.2 and 1.5 pm via 1.3 pm, which is achieved by changing the cycle of short-period growth and the indium composition of the In,Ga,-xAs buffer layer. As shown in Fig. 3.4, emission from the ground state of the ALS dots was observed at 1.34pm, and that from the excited state at 1.24 pm. The FWHM of 28 meV shows a high degree of uniformity in terms of dot size, strain, and composition. Figure 3.8 shows (a) the photoluminescence wavelength and (b) the in-plane diameter of quantum dots evaluated by TEM photography as a function of the number of cycles. The buffer layer was In,,,,Ga,~,,As. By increasing the alternate growth cycle number from 7 to 24, the emission peak wavelength was increased from 1.29 to 1.41 pm. Figure 3.9 shows the photoluminescence wavelength as a function of the buffer-layer indium composition. By increasing the In composition of the buffer layer from 0 (GaAs) to 0.09 (Ino,09Gao,91As), the emission peak wavelength was increased from 1.3 to 1.46 pm at the 12 cycles of InAs/GaAs. Since there was no distinct differences among the In composition in dots measured by EDX, we know that the wavelength variation is due to the change in dot size, as seen in Fig. 3.8(b). Combining the above results concerning the number of supply cycles and the indium composition of the buffer layer, we can control the emission wavelength continuously from 1.2 to 1.5 pm. In order to improve the quantum-dot crystal quality, we optimized various growth conditions such as temperature, sequence, and source supply amount, by checking the photoluminescence emission efficiency. Figure 3.10 shows the photoluminescence intensity versus growth temperature for dots grown at 12 cycles of (InAs),/(GaAs),. The optimized growth temperature was between 440 and 460"C, above and below which the emission intensity rapidly decreased. We also found that the source supply amount should be less than one monolayer. In addition, we adopted a new sequence of In-Ga-As, which provided an emission efficiency more than two times higher than that of the In-As-Ga-As sequence. Under these optimized growth conditions, we made cavity lasers and succeeded in lasing at 80 K, though the emission was from the higher-order excited states (see Chapter 6). We could not achieve lasing from the ground state, because of insufficient optical gain for lasing due to the low area density of the ALS dots. Since the PJE system was not equipped with a gas supply sufficient to fabricate laser devices, we were obliged to grow parts of the laser structure other than the quantum-dot active region using other growth systems. We believe that this process severely damaged the crystal,

3

METALORGANIC VAPORPHASEEPITAXIAL GROWTH

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Number of InAs/GaAs cycles FIG. 3.8. (a) Photoluminescence wavelength at 300 K and (b) in-plane diameter of quantum dots evaluated by a TEM photograph as a function of the number of cycles. The wavelength variation is due to changes in dot size. ((a) Reprinted from Appl. Surface Science, vol. 112, Mukai et a!., "Growth and optical evaluation of InGaAs/GaAs quantum dots self-formed during alternate supply of precursors," pp. 102-109, 1997, with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands; (b) from Mukai et al., 1996.)

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KOHKIMUKAI,MITSURUSUGAWARA, AND MITSURUEGAWA

0

0.05

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Indium composition of buffer layer FIG.3.9. Photoluminescence wavelength at 300 K as a function of the indium composition of the In,Ga,-,As buffer layer. When supplying 12 cycles of alternate supply, the emission wavelength increased from 1.3 to 1.46pm as the indium composition increased from 0 (it., GaAs) to 0.09. By evaluating the reciprocal lattice using X-ray diffraction, it was found that an increase in the indium composition of the buffer layer corresponded to an increase in strain relaxation. (Reprinted from Appl. Sur/ucr Science, vol. 112, Mukai et al., “Growth and optical evaluation of InGaAs/GaAs quantum dots self-formed during alternate supply of precursors,” pp. 102- 109, 1997, with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)

resulting in lasing capability only at low temperature. Unfortunately, we were unable to improve on this, since our research on ALS quantum dots was suspended for almost two years until we acquired another next reactor.

IV. Alternate Supply Growth of InGaAs Dots by In-Ga-As Sequence Supply growth thereafter was performed in a state-of-the-art MOVPE system, a standard type designed by Fujitsu Laboratories and used to produce commercial optical semiconductor devices. The reactor has a vertical configuration with a wide-area inlet consisting of multiple gas injectors (Kondo et al., 1992). The injectors are arranged in a threefold honeycomb configuration so that the inlet gas impinges perpendicularly in the entire area of the substrate. Organometallics, hydrides, and dopants were mixed with the main hydrogen carrier gas in a compact gas-switching

3 METALORGANIC VAPORPHASE EPITAXIAL GROWTH

380

420

460

167

500

Growth temperature (“C) FIG. 3.10. Photoluminescence intensity of ALS dots at 300K as a function of growth temperature. The ALS dots were grown by 12 cycles of (InAs),/(GaAs), alternate supply. Note that this figure is a semi-log plot. (Reprinted from Appl. Surfuee Science, vol. 112, Mukai et al., “Growth and optical evaluation of InGaAs/GaAs quantum dots self-formed during alternate supply of precursors,” pp. 102-109, 1997, with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)

manifold, then divided into subflows and introduced to the injectors. For the gas-handling system, we used a fast switching manifold with a pressurebalanced vent-and-run configuration. Self-limiting growth was observed for InAs and GaAs at common temperatures using TMIDMEA and TMGa. Note that the growth temperature indicated in this section does not agree with the temperature of the PJE reactor in Sections I1 and 111. Comparing temperatures for self-limiting growth in each reactor, a growth temperature of 460°C for the PJE reactor in Sections I1 and 111 corresponds to 520°C for the MOVPE reactor in this section. The most striking change from Sections I1 and 111 was that we adopted the growth sequence In-Ga-As from the beginning. Optimized source supply amounts were 2.0 seconds for TMIDMEA and 0.3 seconds for TMGa, which correspond to 0.5-monolayer In and 0.1-monolayer Ga. The duration of the ASH, supply was fixed at 7 seconds. The H, purge was inserted after the TMGa and ASH, supply for 3 seconds and 0.5 seconds, respectively. The supply rates of TMIDMEA, TMGa, and ASH, sources were 0.04, 4.6, and 40 cc per minute, respectively. A single cycle time of 12.8 seconds provided a very low growth rate of about 0.04 ML per second.

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KOHKIMUKAI, MITSURU SUGAWARA, AND MITSURUEGAWA

In this section, we further explore the growth conditions and processes of ALS quantum dots. We first show that there are two types of ALS quantum dots, depending on the growth temperature and the alternate supply cycle. We then show the multiple-layer growth of quantum dots. OF ALS DOT 1. Two TYPES

The growth procedure was as follows. A 0.25-pm GaAs buffer layer was grown on a (100)-GaAs substrate at 630°C. The growth was then interrupted for 10 minutes and the substrate temperature was lowered under an ASH, atmosphere. Then, after an additional 0.05-pm GaAs buffer layer growth, the In-Ga-As layer was grown by the alternate source supply, followed by the 30-nm GaAs cap layer. The GaAs buffer and cap layers were grown by conventional MOVPE using TEGa and ASH,. The growth pressure was 15 Torr throughout the structure. Figure 3.1 l(a) shows the wavelength of a photoluminescence spectrum peak as a function of the growth temperature for the 18-cycle In-Ga-As supply. The emission wavelength varied considerably between 1.2 and 1.3pm depending on the growth temperature, with a 1.3-pm emission at around 500°C. Figure 3.11(b) shows the peak wavelength of the photoluminescence spectra as a function of the cycle number at various growth temperatures. The relationship between the cycle number and the emission wavelength also varied considerably between the two temperature ranges, that is, the dots grown at 435 and 460°C showed an almost constant emission wavelength, while the emission wavelength of dots grown at 490-510°C increased as the cycle number increased. This suggests that there are two types of ALS dots grown according to different growth processes. The dots with an emission wavelength of 1.2pm are similar to SK dots in that the emission wavelength is independent of the source supply amount. Chapter 2 states that the diameter of SK dots saturates above a critical InAs layer thickness and that the dots are controlled by growth temperature only. Figure 3.12 shows how the photoluminescence spectra vary with cycle number and growth temperature. They were measured at 300 K using a Krf laser with an excitation power density of 60 W/cm2. Using both the peak wavelength and intensity as standards, we classified these photoluminescence spectra into five types. With a small cycle number, a very weak and broad emission was observed, which can be attributed to an emission from the two-dimensional layer prior to three-dimensional nucleation. No dots were observed in the TEM photograph. As the cycle number increased, single peaked and intense emissions appeared at around 1.2 pm (we call this emission type A), followed by double-peaked, intense emissions at around

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1.3 pm (type B). At temperatures above 560"C, a single peaked emission at 1.2 pm weakened (type A'). A double-peaked, intense 1.3-pm emission appeared in the narrow temperature region of 500-550°C. Emissions weakened above 560°C and below 490°C (type B'). The TEM observation revealed the existence of quantum dots in the type A, A', B, and B' regions. Hereafter, the quantum dots characterized by the

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type A and type B photoluminescence spectra are called type A and type B quantum dots respectively. The photoluminescence matrix of Fig. 3.12 tells us that source supply amount and growth temperature govern the formation of ALS quantum dots. For a growth temperature of 5O0-55O0C, type-A quantum dots with a density of about 6 x 10'' c m p 2 (determined from plan-view TEM) h i tially form after two-dimensional growth. The subsequent source supply transforms type-A quantum dots to type-B quantum dots with an identical density. The shorter wavelength peak in the type-B photoluminescence spectra is the second sublevel emission of type-B quantum dots because, as the excitation laser power increases, several peaks appear in the higher energy regime of the second peak (see Fig. 4.11(a) in Chapter 4). The photoluminescence linewidth for type-B quantum dots is narrow, with an FWHM of 30-50meV, while the linewidth for type-A quantum dots is broad, with an FWHM of 80-120 meV. At higher growth temperatures, type-A' quantum dots are transformed into type-B' quantum dots with an increase in source supply. The weak photoluminescence in type-A' and type-B' quantum dots above 560°C can be attributed to their low density, as confirmed by the TEM photograph; the density of type-A' quantum dots at 560°C and 18 cycles was about 2 x 10" cm-2-one-third that of type-A and type-B quantum dots. For growth temperatures below 490"C, type-A quantum dots transform into type-B' quantum dots with weak photoluminescence intensity. An introduction of dislocations reduced the photoluminescence intensity of the type-B' dots below 490°C, as detected by the TEM photograph, although their density was high ( - 1 x 10l1cm-2). Figure 3.13 shows high-resolution cross-sectional TEM images of typical type-A (14-cycle) and type-B ( 1 8-cycle) quantum dots grown at 520°C. Type-A dots have a diameter of 15nm and a height of 5nm. The In For type B the composition detected by EDX is 0.30 (i.e., Ino.30Gao,70As). diameter, height, and In composition were 25 nm, 8 nm, and 0.23, respectively (i.e., Ino,23Gao,77As).Judging from the smaller In composition for type-B dots, the longer photoluminescence wavelength is due to larger dot sizes. We notice that the quantum-well layer surrounding quantum dots in Fig. 3 3 b ) has become thinner like the wetting layer of an SK-mode dot as seen in Fig. 3.13(b). This is because the supply of gallium is reduced from an early amount of almost 1 monolayer to about 0.1 monolayer. The structure of Fig. 3.5(b) can be reproduced in this MOVPE chamber by an In-As-Ga-As sequence (Fig. 3.14(a)) or with an InGaAs layer to cover the dots grown by an In-Ga-As sequence (Fig. 3.14(b)). In the TEM image of Fig. 3.14(a)), the upper and lower borders of the quantum well are not flat, since growth conditions were not yet optimized. In Fig. 3.14(b), type-B quantum dots grown by an In-Ga-As sequence are covered by

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FIG.3.13. Cross-sectlonal TEM images of quantum dots: (a) Type-A quantum dots grown at 520°C with 14 cycles, and (b) Type-B quantum dots grown at 520°C with 18 cycles. The diameter and height of type-A quantum dots are 15 nm and 5 nm, respectively, and those of type-B quantum dots are 25 nm and 8 nm, respectively.

subsequent growth of In,,,Ga,,,As quantum wells. The dots in Figs. 3.14(a) and (b) emit at around 1.3 um (see late news).

2. MULTIPLE-LAYER GROWTH A great deal of effort has been devoted to fabricating quantum-dot arrays that are vertically stacked, vertically aligned, and electronically coupled in the growth direction (Kuan and Iyer, 1991; Yao et al., 1991; Xie et al., 1995; Solomon et al., 1996; Ledentsov et al., 1996a, and b). Vertically aligned arrays can be realized by closely stacking the quantum dots with a

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FIG. 3.14. TEM image of (a) ALS dots grown by an In-As-Ga-As sequence, and (b) ALS dots grown by an In-Ga-As sequence embedded in an In,,,Ga,,,As layer.

spacer-layer thickness comparable to or less than their height. With a simple model showing that new quantum dots tend to nucleate directly above buried quantum dots. Tersoff et al. (1996) demonstrated that the sizes and in-plane spacings of the dots become more uniform with the growth of successive layers. See Chapter 2 for the MBE growth of our closely stacked and columnar SK quantum dots. Here, let us see how multiple stacking proceeds in type-A and type-B quantum dots. We examined triple-layered structures with GaAs spacer layers whose thicknesses varied from 10 nm to 20 nm. The growth temperature was set at 52OoC, which was a favorable temperature for type-B quantum-dot formation. The cycle number of alternate supply was set to 14 and 18 in each layer for both types, respectively. Figures 3.15 and 3.16 show cross-sectional TEM images of triple-layered structures of type-A and type-B quantum dots, respectively; a TEM image

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FIG.3.15. Cross-sectional TEM images of triple-layered structures of type-A quantum dots grown at 520°C with 14 cycles. The GaAs spacers are (a) 20 nm, (b) 15 nm, and (c) 10nm. The single-layer structure of type-A quantum dots is shown in (d).

of a single-layer structure is also shown for comparison. In both types, the well-known vertical alignment of multistacked quantum dots occurred by thinning the GaAs spacer layer, originated by the effect of strain field induced by buried quantum dots. The spacer, at which the vertical alignment occurs, was thicker for type B because the strain field around the buried dots is larger than that for type A due to larger quantum-dot volume.

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FIG. 3.16. Cross-sectional TEM images of triple-layer structures of type-B quantum dots grown at 520'C with 18 cycles. The GaAs spacers are (a) 20nm. (b) 15 nm, and (c) 10nm. The single-layer structure of type-B quantum dots is shown in (d).

Vertically aligned quantum dots became larger toward the upper layers in the triple-layered structure, a phenomenon also observed in SK islands in Chapter 2. The diameter of the dots at the top layer was about 35 nm, while the diameter of those at the bottom layer was about 25 nm. We discovered a unique phenomenon in the multiple-layer growth of type-B quantum dots separated with a thick GaAs spacer. As shown in Fig.

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3.16, in the case of a 20-nm GaAs spacer, the quantum-dot density in the bottom layer increased to about twice that of the single-layer structure, and few dots were formed on the upper layers. For type-A quantum dots, a drastic increase in density did not occur and similar structures were simply repeated. Since the photoluminescence spectrum of the type-B quantum-dot triple-layer structure with a 20-nm spacer was similar to that of the single-layer structure, the newly formed quantum dots in the bottom layer were of type B. At the present stage, we have no idea why the two types of dot display different multiple stacking behavior.

V. The Growth Process We have seen in this chapter that ALS dots are very different from SK islands. Let us assign type-A quantum dots to SK islands because, with what we saw in Chapter 2, they have many common characteristics. The type B ALS dots emit photons with a wavelength of 1.3pm or longer, because of thicker structure than that of the SK islands. The FWHM of the emission spectra is 30-50 meV, which is much smaller than the 80- 120 meV of the SK islands. The ALS dots are buried in a quantum well with clear upper and lower borders against the GaAs. The thickness of the quantum well decreases and becomes like that of a wetting layer in SK-mode growth as the supply amount of group-111 sources is reduced. The area coverage of the dots is 5-lo%, which is definitely smaller than the 20-40% of the SK islands. As the alternate supply cycle increases, dot size and emission wavelength increase, whereas SK islands are almost insensitive to the supply amount of InAs. As seen in Chapter 2, further indium supply after the formation of SK islands increases the islands density and then causes them to coalesce. The photoluminescence matrix of Fig. 3.12 gives us a hint as to how the alternate supply works. We have seen that, as the supply cycle increases the wetting layer and then the SK islands (type A) form, leading to the appearance of ALS dots (type B). The SK islands are self-assembled primarily by the thermodynamical balance between strain energy and surface energy. If we supply only indium after the SK islands (type A) form, dot size will hardly change, as in the case of usual SK growth, since the islands have already reached an equilibrium point. A supply of gallium to the islands decreases the total strain energy, making it possible to add further indium to the islands and thus increase dot size in a perpendicular direction. This process repeats itself until the dots reach another equilibrium point.

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We believe that the growth process is not exactly the same for the early ALS dots grown by supply sequence of In-As-Ga-As and for the recently adopted ALS dots grown by In-Ga-As sequence, with both sequences providing a unique 1.3-pm emission. The plan-view TEM images in Figs. 3.17 and 3.18 indicate how the In source gas contents in a supply cycle affects the structure of the two types of ALS dots. By the increase of source

(b) FIG. 3.17. Plan-view TEM images of the ALS dots grown by 18 cycle of the In-Ga-As supply sequence: (a) the standard case and (b) the case when In source gas contents in a supply cycle was raised by 20%.

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(bi FIG. 3.18. Plan-view TEM images of the ALS dots grown by 12 cycle of the In-As-Ga-As supply sequence: (a) the standard case and (b) the case when In source gas contents in a supply cycle was raised by 100%.

gas contents, the numerical density increases but the diameter does not change for the ALS dots grown by In-Ga-As sequence (Fig. 3.18), while the numerical density does not change but the diameter increases for the ALS dots grown by In-As-GaAs sequence (Fig. 3.17). We don’t know the details in the growth processes, but we suppose that the supply of ASH, after the supply of TMIDMEA makes metal indium on growth surface stable by

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forming semiconductor InAs, affecting the successive process of the ALS dots' formation. We notice that the ALS dots are very similar to the columnar SK dots in Chapter 2 in terms of alternate supply of In and G a sources. The distinct difference is the supply amount, that is, the ALS dots in Section IV supply 0.5 monolayer indium and 0.1 monolayer gallium, while the columnar dots formed by an alternate supply of 3 monolayer GaAs and 0.7 monolayer InAs on SK islands. The role of alternate supply in both types of quantum dots is common, that is, inserting gallium into the structures reduces the total strain energy, leading to larger and more uniform dots. However, we should note that the growth process is not exactly the same for both types of dots since multiple wetting layers are clearly observed in the columnar dot structure but not in the ALS dot structure. We conjecture that reconstruction on surface and/or in bulk may occur during the formation of the ALS dots, in considering with the curious multiple stacking behavior (see Section IV.2). The problem of low density in ALS dots arises from the low source supply amount. If we first form SK islands intentionally by supplying only indium of more than one monolayer (the InAs layer deforms to islands over one-monolayer deposition) and then start an alternate supply, we can increase the dot density, as in the columnar dots grown by MBE. Figure 3.19 shows a trial of the above, photoluminescence spectra at room

0.9

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FIG.3.20. Plan-view TEM image of a sample grown by 10 cycles of an In-Ga-As sequence after supply of 2-monolayer In and As. Quantum dots having an in-plane diameter of 20 nm were observed.

temperature. A sample was grown by 10 cycles of In-Ga-As sequence after a supply of 2-monolayer In and As. A 1.3-pm emission with high emission efficiency was obtained by this hybrid growth method. A plan-view TEM image of the sample indicated the formation of quantum dots, with a diameter of about 20 nm (Fig. 3.20). The numerical density increased slightly (see Fig. 3.5(a)), suggesting that this growth process will solve dot density problems in the future. At the same time, the hybrid method may be able to rectify the fact that multiple layers cannot be formed in ALS dots at present while they can in SK dots. As we progress from closely stacked to columnar dots by continuously thinning the spacer, we must also explore the intermediate regime between ALS dots and columnar dots to expand controllability of size, emission wavelength, numerical density, and emission efficiency. The alternate supply method, which provides great controllability to self-assembling growth, will be a key approach to creating long-wavelength optical devices, such as 1.3-pm quantum-dot lasers.

VI.

Summary

We reviewed the growth characteristics of our original ALS quantum dots, which are grown by an alternate supply of a monolayer or less with a sequence of In-As-Ga-As and In-Ga-As. The dots have a 1.3-pm emission

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and a narrow spectrum linewidth of 28 meV. The quantum dots also acquire great controllability in size and emission wavelength by an adjustment in the cycle number and the composition of buffer layers on which the dots are grown. By comparing the ALS quantum dots with SK islands, especially their spectrum variation with the supply cycle number, we have achieved a picture of the growth process of ALS dots. Optical characterization of ALS dots is presented in Chapter 4.

REFERENCES Kondo, M., Kuramata, A., Fuji, T., Anayama, C., Okazaki, J., Sekiguchi, H.. Tanahashi. T., Yamazaki, S., and Nakajima, K. (1992). J . Cryst. Growth 124, 265. Krost, A., Heinrichsdorff, F., Bimberg, D., Darhuber, A,, and Bauer, G . (1996). Appl. Phys. Lett. 68, 785. Kuan, S. T., and Iyer, S. S. (1991). Appl. Phys. Lett. 59, 2242. Kitamura. M., Nishioka, M., Schur. R.. and Arakawa, Y. (1997). J . Cryst. Growth 170. 563. Ledentsov, N. N., Grundmann, M., Kirstaedter. N., Schmidt, O., Heitz, R., Bohrer, J.. Bimberg, D., Ustinov, M. V., Shchukin, A. V.. Kop'ev, S. P., Alferov, I. Z., Ruvimov, S. S., Kosogov, 0. A., Werner, P., Richter, U., Gosele. U., and Heydenreich, J. (1996a). Solid-State Electron. 40, 785. Ledentsov, N. N., Bohrer, J., Bimberg, D., Kochnev, V. I., Maximov, V. M., Kop'rv, S. P., Alferov, I. Z., Kosogov, 0. A., Ruvimov, S. S., Werner, P., and Gosele, U. (1996b). A p p l . Phys. Lett. 69, 1095. Mukai, K., Ohtsuka, N., Sugawara, M., and Yamazaki, S. (1994). Jpn. J . Appl. Phps. 33. L1710. Mukai, K., Ohtsuka, N., and Sugawara, M. (1996). Jpn. J . Appl. Phys. 35, L262. Mnkai, K., Ohtsuka, N., Shoji, H., and Sugawara, M. (1997). Appl. Surf: Sci. 112, 102. Mon, K., and Usui, S. (1992). Appl. Phys. Lett. 60, 1717. Ohtsuka, N., and Ueda, 0. (1994). In Gus-Phase und Surface Clieinistry in Electronic Materials Processing Vol. 334, 225-22. Ohtsuka, N., and Mukai, K. (1995). Proc,. I n t . Conf 1nP and Related Materials, Hokkaido, 303. Ozeki, M., Mochizuki, K., Ohtsuka, N., and Kodama, K. (1988). Appl. Phys. Lett. 53. 1509. Ozeki, M., Mochizuki, K., Ohtsuka, N..and Kodama, K. (1989). Thin Solid Films 174. 63. Ozeki, M., Ohtsuka, N., Sakuma, Y.. and Kodama, K. (1991). J . Cryst. Growth 107. 102. Ramachandran, R. T., Heitz, R., Chen, P.. and Madhukar, A. (1997a). Appl. Phys. Lett. 70, 640. Ramachandran, R. T., Heitz, R., Kobayashi, P. N., Kalburge, A,, Yu, W., Chen, P., and Madhukar, A. (1997b). J . Cryst. Growth 175/176, 216. Roan, E. J., and Cheng, K. Y. (1991). Appl. Phys. Lett. 59, 2688. Sakuma, Y., Kodama, K., and Ozeki, M. (1988). Jpn. J . Appl. Phys. Lett. 27, L2189. Solomon, S. G., Trezza. A. J., Marshall. F. A.. and Harris. S. J. (1996). Pkys. Rev. Lett. 76, 952. TersoK, J., Teichert. C., and Lagally, G. M. (1996). Phys. Reo. Lett. 76, 1675. Xie, Q., Madhukar, A., Chen, P., and Kobayashi, P. N. (1995). Phys. Reii. Lett. 75, 2542. Yao, Y. J., Andersson, G. T., and Dunlop, L. G. (1991). J . Appl. Phys. 69, 2224.

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SEMICONDUCTORS AND SEMIMETALS, VOL. 60

CHAPTER4

Optical Characterization of Quantum Dots Kohki Mukai and Mitsuru Sugawara OPTICAL SEMICONDUCTOR DEVlCtS LABORATORY FUIITSULABORATORIES LTV ATSUGI.KANAGAWA. JAPAN

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. LIGHTEMISSION FROM DISCRETE ENERGYSTATES . . . . . . . . . . . . . . I, Photoluminescence, Photoluminescence Excitation, and Electroluininescence Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Wafer Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Microprobe Photoluminescence . . . . . . . . . . . . . . . . . . . . 111. CONTROLLABILITY OF QUANTUM CONFINEMENT . . . . . . . . . . . . . . 1. Two Methods of Controlling Quantized Energies . . . . . . . . . . . . 2. Magneto-Optical Spectroscopy . . . . . . . . . . . . . . . . . . . . IV. RADIATIVE EMISSION EFFICIENCY. . . . . . . . . . . . . . . . . . . . V. S U M M A R Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction Optical diagnostics have played a central role in characterizing semiconductor nano-structures. This is because, by probing photons that interact with electrons in nano-structures and by analyzing the optical spectra in time and wavelength regimes, we obtain ample information on the nanostructures’ electronic states, such as energy levels, density of states, band structures, and wave functions, as well as their carrier dynamics, such as relaxation and scattering. Information on strength and resonant wavelength of the photon-electron interaction process itself, such as spontaneous emission, stimulated emission, optical absorption, and various optical nonlinear interactions, is also important in the fabrication of optical devices. Another advantage of optical diagnostics is that, in most cases, we are free from time-consuming sample preparation. In the two decades of quantumwell research, many useful optical diagnostic techniques have been estab183 Copyright i: 1999 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-752169-0 ISSN 0080-R784!99 $30.00

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lished. Photoluminescence is one of the most conventional and useful, providing much information on energy states as well as crystal quality. Among other optical methods are photoluminescence excitation spectroscopy, time-resolved photoluminescence, microprobe optical spectroscopy performed with a microscope or sharpened optical fiber, and magnetooptical measurements. This chapter deals with optical characterization of quantum dots, focusing on the original alternate supply (ALS) quantum dots introduced in Chapter 3 (see also Mukai et al., 1994). The ALS dot is one of four types of quantum dot treated in this volume; the other three are Stranski-Krastanov (SK) dots (Tabuchi et al., 1992), closely stacked dots (Nakata et al., 1997a), and columnar dots (Nakata et al. 1997b; also see Chapter 2). Transmission electron microscope (TEM) photographs of ALS and SK dots are shown in Fig. 4.1. The SK dots are dome-shaped (in another case,

Cross-sectional TEM

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FIG.4.1. Two types of self-assembled dots. The InAs SK dots are grown by MBE or MOVPE via the Stranski-Krastanov (SK) mode under highly strained epitaxy. The InGaAs ALS dots are grown by alternately supplying InAs and GaAs precursors with one or fewer than one monolayer. While SK dots are dome- or pyramid-shaped islands on a growth surface, ALS dots are spherical and surrounded laterally by a quantum-well layer having almost the same thickness as that of the dots.

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pyramid-shaped; Daruka and Barabasi, 1997) islands on a highly latticemismatched growth surface; the ALS dots are spherical and surrounded laterally by a quantum-well layer having almost the same thickness as that of the dots. One of the unique features of ALS dots is that they emit at 1.3 pm-a suitable wavelength for an optical data transmission system. Using diagnostic techniques, we will show the unique properties of ALS dots, such as long emission wavelength, emission spectra with multiple peaks from discrete energy states, harmonic-oscillator type confinement potential, large wavelength tunability between 1.2 and 1.5 pm through size control, and carrier lifetimes through radiative and nonradiative recombinations. We will compare ALS and SK dots, showing the advantages of the alternate supply method.

11. Light Emission from Discrete Energy States

1. PHOTOLUMINESCENCE, PHOTOLUMINESCENCE EXCITATION AND ELECTROLUMINESCENCE SPECTRA Photoluminescence spectra of ALS and SK self-assembled quantum dots at 300K are shown in Fig. 4.2, measured with a Kr’ laser at a low excitation power of 60 W/cm2. The differences between the two are remarkable. The spectrum peak wavelength of the ALS dots is about 1.35pm, which is longer than the 1.12pm of the SK dots. We also see that the spectrum broadening in the ALS dots is less than half that of the SK dots. Figure 4.3 shows the size fluctuation of ALS dots evaluated by plan-view TEM images (see Chapter 3). The diameter distribution follows a Gaussian curve and is comparable to or smaller than that of the SK dots (Leonard et al., 1993; Moison et al., 1993). Since the spectrum consists of the ensemble of an emission line from each quantum dot, the narrower spectrum in ALS dots indicates higher uniformity of dot emission energies and thus higher uniformity of dot size, composition, and strain. Figure 4.4 shows the full width at half maximum (FWHM) of the emission spectra of ALS dots as a function of temperature from 60 to 300 K. Spectra at 4.2, 77, and 300 K are shown in the inset. Temperature-independent spectrum width indicates that the broadening is dominated neither by the thermal carrier distribution nor by homogeneous broadening from phonon scattering, but by structural inhomogeneity. The homogeneous broadening of a single SK dot has been shown by microprobe photoluminescence to be less than 0.1 meV, which is much smaller than the measured spectrum width in Fig. 4.2 (Marzin et al., 1994). Theoretically, the

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Wavelength (pm) FIG. 4.2. Photoluminescence spectra at 300 K of the ALS and SK dots. The ALS dots were buffer layer and capped grown by 18 cycles of In-Ga-As alternate supply on an In,,,,Ga,,,As by an In,,,,Ga,,,,As layer. The emission wavelength of the ALS dots was above 1.3 pm,longer than that of the SK dots. The full width at half maximum (FWHM) of the spectrum was 30 meV in the ALS dots, showing high uniformity. (Reprinted from Appl. Surface Science, vol. 112, Mukai et al., “Growth and optical evaluation of InGaAs/GaAs quantum dots self-formed during alternate supply of precursors,” pp. 102-109, 1997, with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)

homogeneous broadening of each quantum dot, referred to as the natural spectrum linewidth, is controlled by the spontaneous emission rate and carrier scattering rate, as seen in Eq. (1.200) in Chapter 1. (See Eq. (1.210) in Chapter 1 for the spontaneous emission rate and Chapter 7 for the calculation of the scattering rate as a function of the carrier density around dots.) It is yet to be investigated how large the inhomogeneous broadening of a single dot becomes at room temperature and when there are numerous carriers both inside and outside the dots, as is the case with the lasing operation. Note that, in quantum wells, photoluminescence spectra become broad toward high energies as temperature increases, primarily due to the thermal distribution of carriers over continuous energy states. The temperature-independent spectrum width in Fig. 4.4 is one of the factors proving that photoluminescence is caused by the interband transitions between the discrete states in the conduction and valence bands.

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Diameter observed by TEM (nm) FIG. 4.3. Diameter of 106 ALS dots observed in a plan-view TEM image. The distribution follows a Gaussian curve with a standard deviation of 2.9 nm.

The inset of Fig. 4.4 shows additional shoulders at the shorter wavelength regime of the spectra. These were the first sign of excited-level photoemission from self-assembled dots (Mukai et al., 1994). Multiple peak emissions from discrete states can be detected owing to a narrower spectrum linewidth than that of the interlevel separation. Figure 4.5 shows the excitation power dependence of the photoluminescence spectra of ALS dots at 77 K. As the excitation power increased, the second peak appeared, and its intensity increased to exceed the first peak. At the highest power of 306 W/cm2, a shoulder assigned to the third peak appeared. Figure 4.6 compares the photoluminescence spectrum at a rather high excitation intensity of 100 W/cm2, and the photoluminescence excitation spectrum, monitored at the energy of 1.0 eV (the lowest-energy first emission). It was possible to detect two clear resonances exactly at the same energy as those in the photoluminescence spectra. This represents direct proof that the multiple peaks in the photoluminescence spectra are due to the discrete states under three-dimensional quantum confinement. Let us call the first peak in the photoluminescence spectra the ground-state emission and the second peak the first excited-state emission. Figure 4.7 shows the electroluminescence spectra of the ALS dots at 77 K from the p-n junction diode with the dot active region. The area of the

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electrode was 20 x 900 pm'. Since the electric carrier injection can provide many more carriers into dots than the optic excitation can, peaks up to the 5th appeared at the current injection of 400mA. As the injected current increased, the emission intensity of the first and second peaks were almost saturated. Since the emission rate is given by NITsp, where N is the carrier is the spontaneous emission lifetime. the number occupying the state and t S p saturation of the emission intensity is due to the saturation of the carrier number in the corresponding energy state. As low energy states come to be almost occupied by carriers, the relaxation rate into the states decreases due to Pauli blocking, and the emission starts to rise from higher energy states. However, Pauli blocking cannot completely explain the excitation intensity dependence of emission spectra in Fig. 4.7. We should note that, at lOmA, the second and third emissions had appeared though the first emission intensity was less than its maximum. The carrier number, N , of each state, governing the emission intensity, is determined by the balance among the rates of various processes, such as spontaneous emission at the state, relaxation into the state, and thermal emission from the state. In the ALS dots with an energy separation of 50-80 meV-much higher than the thermal energy at 77 K-the thermal emission rate is negligible and the

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experimental emission spectrum with multiple peaks even at low excitation indicates that the relaxation rate is comparable to the spontaneous emission rate. This is a phonon bottleneck (discussed in detail in Chapter 5). One notices immediately about this bottleneck that the emission spectrum at IOmA in Fig.4.7 cannot be reproduced by a calculation that simply combines the Fermi-Dirac distribution at 77 K and the density of states of the quantum dots. Due to the phonon bottleneck effect, N does not completely reach 2 N , to give maximum intensity. Three features of the emission spectra of ALS dots are observed in Fig. 4.7. First, the higher the peak order, the larger the maximum peak intensity. Second, the peak energies were constant during the increase of injected current, supporting our belief that the observed peaks correspond to sublevels in the dots. Third, the peak intervals between the neighboring emission lines are almost uniform -a characteristic of harmonic-oscillatortype quantum confinement. The confinement potential is determined by the magneto-optical measurements described in Section 111.

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Energy (eV) FIG.4.6. Photoluminescence excitation spatrum at 4.2 K of the sample grown by 18 cycles of In-Ga-As supply. The sample was excited by the monochromated light of a Halogen lamp, and a Ge detector was used to detect luminescence from the sample surface at the energy of 1.0 eV, that is, the lowest-energy first emission. Two clear resonances could be detected exactly at the same energy as that in the photoluminescence spectra (from Mukai et al., 1996c. Copyright 1996 by the American Physical Society).

2. WAFERMAPPING Figure 4.8 shows the microprobe photoluminescence mapping of an ALS quantum-dot wafer at room temperature: (a) the energy separation between the ground-state (first) and the first excited-state (second) emission peaks; (b) the ground-state emission energy; and (c) the FWHM of the groundstate emission spectrum. The correction of luminescence from the sample surface was restricted to the area measuring 2 x 2 pm2 using a crossing slit. The sample was grown by 14 cycles of In-Ga-As supply on a 2-inch (100) GaAs wafer in a state-of-the-art MOVPE chamber (see Chapter 3 for details on growth). During growth, the wafer was rotated at the rate of 30rpm, which caused the distribution of the three diagnostic items to have a roughly rotational symmetry around the wafer center. By comparing the three maps, we can deduce what caused the observed patterns. The energy separation of Fig. 4.8(a) is in the 85- to 95-meV range and is almost uniform except at the wafer edge. The ground-state emission energy in Fig. 4.8(b) was the largest at about 1 inch from the wafer center.

4 OPTICAL CHARACTERIZATION OF QUANTUM DOTS

0.9

1.o

1.1

1.2

191

1.3

Wavelength (pm) FIG. 4.7. Electroluminescence spectra of ALS dots grown by 18 cycles of In-Ga-As alternate supply. The luminescence parallel to the surface was dispersed and detected by an InGaAs photo-multi detector quenched to - 70°C using a conventional lock-in technique. Electrode size was 20 x 900prn'. Up to the fifth emission peaks appeared as the injected current increased (from Mukai et al., 1996b).

The FWHM in Fig. 4.8(c) had a distribution similar to that of the ground-state emission energy of (b) -the higher the ground-state energy, the narrower the spectrum width. To understand the patterns, we note two factors causing the observed distributions: the dot size and the indium composition of the dots. As the quantum-dot size decreases, all three items of Fig. 4.8- the energy separation, the ground-state emission energy, and the spectrum width -increase. As the indium composition decreases, the ground-state emission energy increases but the energy separation hardly increases. The observed maps are explained as follows. First, energy separation in Fig. 4.8(a) suggests the good uniformity of dot size all over the wafer size. Second, the high-energy region off the wafer center in Fig. 4.8(b) has the smallest indium composition but does not have the smallest dot size. Third, the narrowest spectrum broadening off the wafer center in Fig. 4.8(c) is due to the uniform indium composition. This indicates that spectrum broadening is caused not only by size fluctuation but also by compositional fluctuation of indium among each quantum dot.

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KOHKIMUKAI AND MITSURUSUGAWARA

0

90-95meV 85 - 90 80 - 85 70-80

Energy separation

0

095-O96eV 0 94 - 0 95

093-094

Ground-level energy

0

a

40-50me~ 50-60

unmeasurable

FWHM of ground level

FIG.4.8. Area mapping of the photoluminescence of an ALS quantum-dot wafer at room t e w r a t u r e : (a) energy separation between the ground-state (first) and the first excited-state (second) emission peaks; (b) ground-state emission energy: and (c) FWHM of the ground-state emission spectrum. The probe size was 2 p m . The scan step was 2mm in both x and y directions. The sample was excited by an Ar+ laser at 514.5 nm, and luminescence from the sample surface was monochromed by a 50-cm rnonochrorneter and detected by a multichannel photodetector.

3. MICROPROBE PHOTOLUMINESCENCE Increasing the spatial resolution of the optical probe is a major target in the development of optical diagnostic techniques. It enables us to obtain signals from a small number of dots, leading to an understanding of quantum-dot optical properties that was heretofore unobtainable through averaged signals over a large number of dots by macroscopic measurements (Brunner et al., 1992). Microscope photoluminescence that focuses the excitation laser beam via an objective lens to about 1-pm diameter on a sample surface is already a popular technique, and its experimental setup is commercially available. Higher spatial resolution is obtained via simple sample processing that opens a window in a metal mask covering the surface through which the photoluminescence can be measured. Sharp emission lines caused by a single dot or a small number of dots can be detected, and the linewidth -homogeneous broadening of a single-dot emission line has been evaluated by several researchers (see Marzin et al., 1994; Hessman et al., 1996).

4 OPTICAL CHARACTERIZATION OF QUANTUM DOTS

193

FIG. 4.9. Near-field optical microscope. Samples were set in the cryostat and cooled to 5 K. With a wavelength of 633nm, the excitation beam went from the laser diode through a sharpened optical fiber to the sample surface. Luminescence from the surface was collected by the fiber probe. The tip was controlled in close proximity to the sample by the shear-force feedback technique. The luminescence was dispersed by a monochrometer, and the photon signal was counted with a n InGaAs photo-multi detector kept at -8O'C.

The other approach to microprobe spectroscopy is near-field optical microscopy (fiber probe microscopy) (Durig et al., 1986; Betzig et al., 1991), which involves collecting near-field light. Its advantage is that spatial resolution is not limited by the wavelength of the excitation laser; its drawback is weak signal intensity. With the shear-force feedback technique, high resolution optical imaging can be obtained by tracing the surface structure of samples (Betzig et al., 1993; Ambrose et al., 1994; Grober et al., 1994; Ghoemi et al., 1995). We performed microprobe photoluminescence spectroscopy of ALS dots using the near-field scanning optical microscopy system (NSOM). Our experimental equipment is illustrated in Fig. 4.9. The excitation beam of a wavelength of 633 nm from the laser diode went through an optical fiber to the sample surface, and luminescence from the surface was collected by the fiber probe. The luminescence was monochromed by a monochrometer with a resolution of 1 nm. The tip was kept in close proximity to the sample (- 10 nm) by applying shear-force feedback (Betzig et al., 1992). The fiber probe resolution was restricted to 1 pm, because of the tradeoff between spatial resolution and detection sensitivity (Saiki et al., 1996). The number of the dots was estimated to be about 200 in a 1-pm spot. Figure 4.10 shows the luminescence spectrum of the ALS dots by the NSOM system with an excitation power of 90 W/cm2. Spectra of inacroscopic photoluminescence with a 300-,um$excitation laser spot are superim-

-

194

KOHKIMUKAIAND MITSURUSUGAWARA

c

0 0 d Y

a

.

NSOM

90 W/cm2

:

r . . . . l . . . . l . . . . l . . . . i 0.95 1.oo I .05 1.10 1.15 Energy (eV) FIG.4.10. Photoluminescence spectra of the ALS dots by the near-field optical microscope. The excitation power was estimated to be 90 W/cm2. The spectra of macro-photoluminescence is shown by a dashed line. The two sets of multiple emission peaks in the NSOM spectra correspond to the ground-state and the first excited-state emission.

posed by a dashed line. The two sets of emission spectra with fine structures denoted as S, and S,, correspond to the quantum dots' ground state and first excited state, respectively. Note that S, and S, are well separated and that almost no emission was observed in between. This is because the smaller probing area reduced the inhomogeneous spectrum broadening. However, the emission spectrum of S, and S, did not become as narrow as we expected, indicating that inhomogeneity of dots had already occurred in the area smaller than 1 pm. Inhomogeneity in such a small area cannot be caused by nonuniformity in growth conditions, such as wafer temperature distribution and source gas streamline. The fine structures on S, and S, are reproduced by repeated measurements and can be attributed to emission from one or a few dots. Sharp peaks, denoted by the common small letter with the subscripts 1 and 2, might be the ground-state and the excited-state emission of the same dots. Figure 4.1 l(a) shows the excitation-power dependence of the NSOM spectra. Comparing that with the macroscopic photoluminescence spectra of Fig. 4.5, we see that the emission from each state is sharper and up to the seventh emission is resolved, due to the higher spatial resolution. We found that the higher the sublevel order, the wider the peak width. Thus, the

4 OPTICAL CHARACTERIZATION OF QUANTUM DOTS

0.9

1

1.1

1.2

1.3

1.4

1.5

195

1.6

Energy (eV)

0.95

1.oo

1.05

Energy (eV) FIG.4.1 1. (a) Photoluminescence spectra of the ALS dots by a near-field optical microscope at various excitation-power densities. As the excitation power increased, a higher-energy emission peak was clearly discernable. (b) NSOM spectra around the ground-state emission. An additional emission appeared at the lower-energy side of the peak at high excitation (from Mukai et al., 399713).

196

KOHKIMUKAIAND MITSURUSUGAWARA

higher-ordered peaks were not well isolated, perhaps because the energy interval was not uniform in dot ensembles in the area; the discrepancy in the interval enhances the peak broadening at the excited sublevels. Another reason might be that an additional emission appeared at the lower-energy side in each peak. The spectrum around the ground-state emission is magnified in Fig. 4.11(b). As the excitation power was increased from 90W/cm2 to 9kW/cmz, the peak emission intensity of the ground state decreased and the lower-energy emission rose. This red shift of emission might be due to the many-body effect in quantum dots (Sugawara, 1996).

111. Controllability of Quantum Confinement

The energy states (e.g., quantized energies and density of states) and optical characteristics of quantum nano-structures can be artificially controlled by changing the size of the structures. In quantum wells, well width and strain are designed to match target devices using well-established k . p band calculation methods (for example, see Chuang, 1995) and the designed structures are precisely fabricated in the atomic scale by state-of-the-art epitaxial growth techniques like MOVPE and MBE. The ability to control quantum-dot structures is not yet as strong as it could be in spite of its significance for practical device applications. In self-assembled InAs SK dots on GaAs substrates, the size and emission wavelength can be controlled only within a narrow range by changing the growth temperature (Chapter 2). Changing the supply amount of sources has a negligible effect because the assembling process is driven toward a thermodynamical equilibrium point. In this section, we will introduce two methods of controlling the size and the emission wavelength of InGaAs ALS quantum dots over a wide range of 1.2 to 1.5pm. Then we will evaluate the quantum-dot confinement potential by magneto-optical measurements.

OF CONTROLLING QUANTIZED ENERGIES 1. TWO METHODS

The first method of controlling the emission wavelength of quantum dots is to change the number of supply cycles in the alternate supply growth (See Chapter 3 for the ALS growth process). Figure 4.12 shows the 4.2-K photoluminescence spectra of ALS quantum dots at various cycle number of the In-As-Ga-As supply cycle. At cycle 7, the dots were assembled to release the strain energy accumulated by the lattice mismatch between InAs and GaAs. As the number of cycles increased from 7 to 30, the peak

4 OPTICAL CHARACTERIZATION OF QUANTUM DOTS

197

4.2 K

1.o

1.1

1.2

I .3

I .4

Wavelength (pm) FIG. 4.12. Photoluminescence spectra of the ALS dots grown by various cycles of In-AsGa-As alternate supply. As the number of cycles increased, the peak wavelength shifted from 1.17 to 1.3 pm. The second peaks were observed at the left side of the first peak. The maximum emission intensity was obtained by the sample grown by 9 cycles (from Mukai et al., 1996a).

wavelength shifted from 1.17 to 1.3pm. We observed weak peaks or shoulders (indicated by arrows) that had higher emission energies than the first peak had. In the 7-cycle sample, the second peak was hardly observed, probably due to the low optical quality. As the number of cycles increased, the second peak moved toward the first peak, and in the 30-cycle sample the two merged. Plan-view TEM images of the samples in Fig. 4.13 show that, as the supply cycles increased, so did the diameter of the dots. As summarized in Fig. 4.14, the cycle increase enlarges the diameter and reduces the emission energy and the energy separation between the two peaks. If the first peak is assigned to the ground-state emission and the second peak to the exited-state emission, as in Section IT, the results in Fig.

198

KOHKIMUKAI AND MITSURUSUGAWARA

(C)

-

40nm

FIG.4.13. Plan-view TEM images of ALS dots grown with (a) 9 cycles, (b) 12 cycles, and (c) 24 cycles. The dot diameter increased as the number of cycles increased.

4.14 go well with the quantum-confinement concept of energy increase due to confinement. The maximum emission intensity and the narrowest spectrum width were achieved at the same time at 9 cycles, showing that the alternate supply method controls not only the size but also the crystal quality. The second way to control emission wavelength is to change the lattice constant of the layer on which the dots are grown. This method is based on the idea that formation of the self-assembled three-dimensional islands is motivated by the release of accumulated strain energy, and that the size of the islands increases as the lattice mismatch between the substrate and the supplied materials decreases. We grew 500-nm In,Ga, -,As buffer layers with the composition of x = 0 to 0.09 on GaAs substrates. A reciprocal space map of an X-ray diffraction showed that up to 50% strain relaxation occurred in the In,Ga, - ,As buffer layer and that the lattice constant of the buffer layer increased with an increase of x due to the relaxation of strain. Figure 4.15 shows that, as the indium composition of the buffer layer increased from 0 (GaAs) to 0.09 (Ino,09Gao,91As),the emission wavelength increased from 1.3 to 1.46 pm when supplying 12 cycles. TEM measurements

199

4 OPTICAL CHARACTERIZATION OF QUANTUM DOTS

- 35 1.05 -

n

%

- 30

W

x

0

1st 1.00

~

25

0.95l' '

5

4.2 K

lI "

'

I

'

10

' I

" "

15

I '

"

* I

20

" "

25

'

I

30

"

- 20 'I 35

(InAs)/(GaAs) supply cycle number FIG. 4.14. Energies of the first and second peaks in the photoluminescence spectra and dot size observed by TEM as a function of the supply cycle number during ALS quantum-dot growth (from Mukai et al., 1996b).

1.45 h

F----A IntAsfGdAs x 12 cycles

E

3.

W

1.35 l1!

1.3

i/ c

0

--I

0.05

0.1

Indium composition of buffer layer FIG. 4.15. Emission wavelength as a function of the indium composition of an InGaAs buffer layer. With 12 cycles of In-As-Cia-As alternate supply, the emission wavelength increased from 1.3 to 1.46pm as the indium composition increased from 0 (i.e., GaAs) to 0.09 (from Ohtsuka and Mukai, 1995).

200

KOHKIMUKAIAND MITSURUSUGAWARA

Indium composition of InGaAs buffer layer FIG.4.16. In-plane dot diameter observed by plan-view TEM and dot composition measured by EDX as a function of the indium composition of the buffer layer. Composition was determined by calibrating the signal intensity, using the intensity of the lattice-matched In,.,,Ga, ,,As/InP as a reference. The indium composition was between 0.4 and 0.6 in these samples and did not obviously depend on the buffer layer's composition. The diameter increased as the indium composition of the buffer layer increased.

show that the diameter of the ALS dots increased from 20 nm to 35 nm as the indium composition of the buffer layer increased (Fig. 4.16). Energy dispersive X-ray microanalysis (EDX) determined that the composition of the dots was between 0.4 and 0.6 and did not obviously depend on the composition of the buffer layer. By combining two control methods just described, we can control the emission wavelength to the ALS dots from 1.2 to 1.5pm. The buffer layer can be also applied to tune the size and emission wavelength of ordinary SK dots.

2.

MAGNETO-OPTICAL SPECTROSCOPY

Exciton diamagnetic shifts are powerful tools for evaluating not only the reduced effectivemass but also the magnitude of potential affecting excitons. The magnetic field confines excitons in the plane perpendicular to the field and increases their energy. We can determine the magnitude of other

4 OPTICAL CHARACTERIZATION OF QUANTUM DOTS

201

competing confinement potentials by evaluating the diamagnetic energy shifts. In studying the diamagnetic shifts of luminescence from 1.3-pmemitting InGaAsP/InP quantum wells and the ALS dots, we detected the confinement potentials for localized excitons in the quantum wells and quantum dots. Magnetic fields were applied perpendicularly to the wafer using a superconducting magnet immersed in liquid helium. The sample was placed near the center of this magnet, which can generate a magnetic field of up to 14 Tesla. A multiline Ar' laser beam was fed to the sample through an optical fiber bundle. The photoluminescence from the sample surface was led to a 32-cm monochrometer through the optical fiber and detected by a cooled InGaAs photomultiplier using a conventional lock-in technique. Figure 4.17(a) shows measured magnetic-field-induced energy shifts of the emission spectra of dot samples grown by various alternate supply cycles. The diamagnetic shifts of the exciton emission at 1.3 pm in InGaAsP/InP quantum wells are superimposed as a reference. The shifts in dots were smaller than those in quantum wells due to the in-plane confinement potential, and they increased with the cycle number. In 9-cycle samples the shift was negligible. By simulating the energy shifts using the harmonic oscillator-type confinement potential introduced in Chapter 1, we evaluated the in-plane confinement potential of quantum dots. The effective-mass equation for this problem is Eq. (1.214) in Chapter 1 with the confinement potential of rn,*l1w,2/2.Solid lines in Fig. 4.18 show the calculated diamagnetic shifts using the confinement frequency of 0 , as a parameter. The reduced effective mass of m,*" = 0.04~1,and the static dielectric constant of c = 1 3 . 9 ~were ~ used (Sugawara et al., 1993). Comparing the calculated shifts with the measured ones, we obtained w, = 7 x IOl3 s - l for 24 cycles, 1 x 1014s-' for 18 cycles, 1.3 x 1014 s-' for 12 cycles, and 4 x l O I 4 s - l for 9 cycles. As the cycle number increases, the confinement potential spreads. Figure 4.17(b) compares the confinement potential estimated by TEM and EDX (dashed line) with that estimated by magneto-optical evaluation (solid line) for the sample grown by 12 cycles. The origin of the x-axis was set at the center of the dot. The two estimations agree.

IV. Radiative Emission Efficiency Quantum-dot laser performance -for example, threshold currents and external quantum efficiencies-are quite sensitive to the crystal quality of dots, as seen in the simulation in Chapter 1. This is because nonradiative centers, if produced during the self-assembling process, consume carriers

202

KOHKIMUKAIAND MITSURU SUGAWARA

0

50

100

150

Square of magnetic field (T2)

In-plane distance (nm) FIG.4.17. (a) Measured and calculated magnetic-field-induced energy shifts as a function of the square of the magnetic field. Diamagnetic shifts of exciton resonance in InGaAsP quantum wells emitting at 1.3pm are superimposed as a reference. The shifts in quantum dots were smaller than those in quantum wells and increased with the number of supply cycles. Lateral quantum-confinement potentials were determined by fitting the calculation (solid lines) to the measured shifts. (b) Confinement potential estimated by TEM and EDX (dashed line) and that estimated by magneto-optical evaluation (solid line) for the sample grown using 12 cycles. The origin of the x-axis was set at the center of the dot. The two results agree well. ((a) Reprinted from Appl. Surfure Science, vol. 112, Mukai et a]., “Growth and optical evaluation of InGaAs/GaAs quantum dots self-formed during alternate supply of precursors,” pp. 102-109. 1997, with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands; (b) from Mukai et al.. 1994.)

4 OPTICAL CHARACTERIZATION OF QUANTUM DOTS

203

injected into the laser active region. We must pay attention to the quality of both inside dots and outside dots, since the retarded carrier relaxation peculiar to quantum dots enhances the opportunity for carriers to be killed before they relax. Here, we evaluate the nonradiative channel in quantumdot crystals by measuring time-resolved photoluminescence decay and photoluminescence intensity as a function of temperature. Recombination lifetimes of the discrete emission spectra from ALS dots are shown in Fig. 4.18. (See Chapter 5 for details on time-resolved photoluminescence measurements.) The measured lifetimes, T " , are almost independent of the emission level and are about 0.8 to 1.5 nanoseconds, which we judge to be long enough for laser application. The lifetime is slightly shorter than the calculated spontaneous emission lifetime of z S p =2.8 ns using Eq. (1.210) of Chapter 1, with the exciton effect (the Coulomb enhancement effect) neglected. We are not sure whether the slightly shorter experimental lifetime is due to a simple error caused by missing factors in the theory or to uncertain parameters in the calculation or to nonradiative process in the measurements. If we estimate the nonradiative lifetime, z,,, using the relationship of T,- = t,;’ + T,;' and z, = 0.8 - 1.5 ns, we get T,, = 0.6 - 1.0ns. The measured lifetimes are almost independent of temperature up to 300 K because of the discrete quantized levels that prevent

0

+0

0

4

54 0

10-'O r

'f

2nd

.$

+5th

A 3rd

0

50

100

150

200

250

300

Temperature (K) FIG. 4.18. Recombination lifetimes of the five emission peaks in the ALS dots measured by time-resolved photoluminescence. The lifetimes were about one nanosecond and almost independent of temperature up to 300 K .

204

KOHKIMUKAIAND MITSURUSUGAWARA

exciton thermal distribution, which are peculiar to quantum dots (see Chapter 1). Since T,, normally depends on temperature, we assume that the nonradiative process did not influence the recombination lifetime. Figure 4.19 shows the photoluminescence intensity of the ground-state emission of ALS dots as a function of temperature. The intensity was normalized by the value at 4 K. Reported photoluminescence intensities of the SK dots grown by MBE are also shown as references. From 4 K to 300 K, the photoluminescence intensity of the ALS dots decreased monotonously by about one order of magnitude. This decrease is almost equal to, or even smaller than, that of the quantum wells (Mirin et al., 1995). The photoluminescence intensity of the SK dots decreased by almost three orders of magnitude. For the SK dots, there seems to be a critical temperature between 150 and 200 K, where the slope inclination changes,

- Calculation

1.1

1.3

4

1.5

FIG.4.19. Photoluminescence intensity of the ALS dots as a function of temperature. The intensity was normalized by the value at a low temperature. Excitation was done by a 647.1-nm Kr’ ion laser. Reported intensities of the SK dots grown by MBE are also shown as references. The decrease in intensity of the ALS dots following a temperature increase was monotonous and about one order of magnitude at 300K. Photoluminescence intensity of the SK dots decreased by almost three orders of magnitude, and there seemed to be a critical temperature between 150 and 200 K, where the slope inclination changed. Solid and dashed lines indicate the results of fitting. (Reprinted from Appl. Surfhce Science, vol. 112, Mukai et al., “Growth and optical evaluation of InGaAs/GaAs quantum dots self-formed during alternate supply of precursors,” pp. 102- 109, 1997, with kind permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)

4 OPTICAL CHARACTERIZATION OF QUANTUM DOTS

205

suggesting that there are at least two sources for the reduction in photoluminescence intensity. The temperature independence of the recombination lifetimes of Fig. 4.19 tells us where the nonradiative centers are located: The centers outside the dots degrade emission efficiency as temperature increases. Since the excitation wavelength was 647.1 nm, carriers were initially excited outside the dots and diffused into the dots to emit inside them. Assuming nonradiative centers outside the dots, we calculated the temperature dependence of photoluminescence intensity. We neglected the radiative recombination outside the dots, since no emission was observed from the GaAs substrate and the In,,, ,Ga,,.,As layer surrounding the dots. The photoluminescence intensity can thus be described using

where G is the carrier generation rate and zd is the carrier relaxation time from outside to the ground state of the dots. Since only the emission from the ground state was observed, we included the intersubband carrier relaxation time in z d . The lifetime of zz:'(T) is the nonradiative recombination lifetime outside the dots and is written as

(Jm),

where N is the trap density, V(T) is the carrier thermal velocity and oout( T ) is the capture cross-section. We can describe the cross-section using the activation energy of Q as (T~"'(T) = CT x exp( - Q/kT)

(4.3)

By a change of two parameters, N x o and Q, in Eqs. (4.1) through (4.3), the measured emission intensity was simulated by the calculation, as shown by the solid line for the ALS dots and the dashed line for the SK dots in Fig. 4.19. The carrier relaxation time was set at zd = 1 ns from the results of time-resolved measurements (see Chapter 5). The parameter set of [ N x o,Q] = [102cm-', 35meVl explains the results of the ALS dots, whereas the combination of [lo2 cm- 35 meV] and [lo8 cm-', 280meVl describes the results of the MBE-grown dots. The former set is common with the ALS dots; the latter set explains the large bending over 150K. If we use zd = 100 ps, the values of N x o decrease by one order of magnitude. Table 4.1 lists the known parameters of nonradiative centers in the InGaAs system. The parameter set of the ALS dots in calculation is close to

206

KOHKIMUKAIAND MITSURUSUGAWARA TABLE 4.1

OF NONRADIATIVE CENTERS IN InGaAs MATERIALS. E IS THE ACTIVATION REPORTED PARAMETERS EMISSION RATE,B IS THE CAPTURE CROSS-SECTION, Q IS THE ENERGYFOR ne THERMAL ACTIVATION ENERGY FOR THE CAPTURE CROSS-SECTION, AND N IS THE TRAP DENSITY. “ V P E IS AN ABBREVIATION FOR VAPORPHASE EPITAXY. (From Mukai et al., 1997a.)

E (W a’ b2 C‘

d’ el.2

f2.3

g4 h4 i4

0.82 0.54 0.21 0.74 0.53 0.17 0.83 0.76 0.68

B

(cm2)

1.6E-16 3.1E-16 5.9E-14 1.6E-13 5E-15 2E-12 1E-14 1E-15 1E-16

Q (meW

-

80

80

40

N(~m-~) 2.6E + 13 1.OE + 13 7E + 12 2.1E + 14 1.4E + 15 3E + 15

Growth Method MBE MBE MBE MBE MBE, VPE MBE, VPE VPE VPE VPE

‘Bhattacharya, P. K., and Dhar, S. (1988). Semicond. and Semimeta. 26. 20gawa, M., Hongo, S., Watanabe, Y., Sano, N., Katoh, H., Nakayama, M., Ishida, K., and Shirafuji, J. Sixth Record of Alloy Semicond. Phys. and Electro. Symposium, Hakone, Japan 1987. 30gura, M., Mizuta, M., Onaka, K., and Kukimoto, H. (1983). J p n . J . Appl. Phys. 22, 1502. 4Mircea, A,, Mitonneau, A,, Hallais, J., and Jaros, M. (1997). Phys. Reu. B. B16, 3665.

the sets in Table 4.1. Remarkably the parameter set of the MBE-grown dots is much different from the sets in Table 4.1, which suggests that the point defects outside the dots cannot explain the MBE-grown dots’ near-threeorder decrease. Other origins, such as the network of dislocation lines or the nonradiative center inside the dots should be assumed to explain the large parameters. The high emission efficiency of ALS dots can be attributed to their unique growth process. One possible reason for their better optical quality is that the quality of their interface is higher. The ALS dots are surrounded laterally by a quantum-well layer, and the interface around them is ambiguous (see Fig. 4.1). This may be because the atoms around the dots reconstruct the interface during the growth, given that ALS dots are supposedly formed through the repetition of an equilibrium point and a non-equilibrium point of thermodynamical balance between strain energy and surface energy (see Chapter 3). On the other hand, the interface of the SK dots is abrupt, as it is exactly the border between the growth islands and the overgrown layer. Another possible reason for the better optical quality of the ALS dots is that they are in-situ annealed due to the low growth rate of the alternate supply of precursors. The growth rate was about 1

4 OPTICAL CHARACTERIZATION OF QUANTUM DOTS

207

monolayer per 10 seconds for the alternate supply -significantly smaller than that of MBE. The growth temperature of ALS dots is lower than the temperature where arsenic desorption occurs (Sakuma et al., 1993), and some defects may have time enough to go out from the surface during the slow growth.

V. Summary The optical characteristics of ALS dots were compared with those of SK dots. Emission from discrete levels caused by three-dimensional quantum confinement was observed using photoluminescence, electroluminescence, and photoluminescence excitation. The photoluminescence mapping of a wafer revealed the compositional fluctuation of indium among quantum dots. Microprobe photoluminescence revealed the fine structure of emission. Quantum confinement potential was controlled by changing the number of supply cycles and the composition of the buffer layer, and was evaluated by diamagnetic energy shifts. Time-resolved photoluminescence and photoluminescence intensity as a function of temperature were studied to evaluate the nonradiative recombination channel inside and outside the dots. In comparing ALS dots with SK dots, the 1.3-pm emission wavelength is the most remarkable advantage of the ALS dots, as it means that they can be applied to lasers for optical telecommunication systems. Good controllability of the wavelength of ALS dots is also significant in practical device manufacturing. The ALS dots achieved a spectral width of 30 meV -less than half that of the SK dot -and have better emission efficiency. ALS dots have the drawback of low density and insufficient multiple stacking (Chapter 3) which prevent the realization of high-performance 1.3-pm lasers. Research on eliminating these drawbacks is not being carried out. Optical evaluation suggests that self-assembled dots achieve discrete-state density and results in unique characteristics not available before. We are confident that quantum-dot devices will have a significant impact on optoelectronics applications in the near future. Besides its intrinsic importance, optical characterization is indispensable for the development of practical devices. One of the issues we should pursue is the carrier relaxation problem discussed in Chapter 1 as it relates to quantum-dot laser performance. This topic will be treated in Chapter 5.

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REFERENCES Ambrose, P. W.. Goodwin, M. P., Martin, C . J., and Keller, A. R. (1994). Phys. Reu. Lett. 72, 160. Bimberg, D., Ledentsov, N. N., Kirstaedter, N., Schmidt, O., Grundmann, M., Ustinov, M., V., Egorov, Y. A., Zhukov, E. A., Maximov, V. M., Kop’ev, S. P., and Alferov, I., 2 . (1995). Ext. Abs. 1995 Int. Con$ Solid Slate Devices and Materials, Osaka. Brunner, K., Bockelmann, U., Abstreiter, G., Walther, M., Bohm, G., Trankle, G., and Weimann, G. (1992). Pliys. Rev. Leu. 69, 3218. Betzig, E., Trautman, K. J., Harris, D. T., Weiner, S. J., and Kostelak, L. R. (1991). Science 251, 1468. Betzig, E., Finn, L. P., and Weiner, S. J. (1992). Appl. Phys. Lett. 60, 2484. Betzig, E., and Chicheser, J. R. (1993). Science 262, 1422. Chuang, S. L. (1995. Physics ofOptoelectronics Devices, John Wiley & Sons, New York. Daruka, I., and Barabasi, A.-L. (1997). Phys. Rev. Lett. 79, 3708. Diirig, U., Pohl, W. D., and Rohner, F. (1986) J . Appl. Phys. 59, 3318. Grober, D. R., Harris, D. T., Trautrnann. K. J., Betzig, E., Wegscheider, W., Pfeiffer, L., and West, K. (1994). Appl. Phys. Lett. 64. 1421. Ghaemi, F. H., Goldberg, B. B., Cates, C.. Wang, D. P., Sotomayor-Torres, M. C., Fritze, M., and Nurmikko, A. (1995). Superlattices and Microstructures 17, 15. Hessman, D., Castrillo, P., Pistol, E. M., Pryor, C., and Samuelson, L. (1996). Appl. Phys. Lett. 69, 749. Leonard, D., Kishnamurthy, M., Reaves, M. C., Denbaars, P. S., and Petroff, M. P. (1993). Appl. Phys. Lett. 63, 3203. Marzin, Y. J., Gerard, M. J., Izrael, A,. Barrier, D., Bastard, G. (1994). Phys. Rev. Lett. 73, 716. Moison, M. J., Houzay, F., Barthe, F., and Leprince, L. (1993). Appl. Phys. Lett. 64, 196. Mukai, K., Ohtsuka, N., Sugawara, M., and Yarnazaki, S. (1994). Jpn. J . Appl. Phys. 33, L1710. Mukai, K., Ohtsuka, N., and Sugawara, M. (1996a). Jpn. J . Appl. Phys. 35, L262. Mukai, K., Shoji, H., Ohtsuka, N., and Sugawara, M. (1996b). Appl. Phys. Lett. 68, 3013. Mukai, K., Ohtsuka, N., Shoji, H., and Sugawara, M. (1996~).Phys. Rev. B. 54, R5243. Mukai, K., Ohtsuka, N., Shoji, H., and Sugawara. M. (1997a). Appl. Surf Sci. 112, 102. Mukai, K., Ohtsuka, N., and Sugawara. M. (1997b). Ext. Abs. 58th Autumn Meet. Japan Society of Applied Physics, 4pS-8 (in Japanese). Mirin, R. P., Ibbetson, P. J., Nishi, K., Gossard, C. A., and Bowers, E. J. (1995). Appl. Phys. Lett. 67, 3795. Nakata, Y., Sugiyama, Y., Futatsugi, T.. and Yokoyama, N. (1997a). J . Crystal Growth 175/176, 713. Nakata, Y., Sugiyama, Y., Futatsugi, T., Mukai, K., Shoji, H., Sugawara, M., Ishikawa, H., and Yokoyama, N. (1997b). Ext. Abs. 58th Autumn Meet Japan Society of Applied Physics, 2pM-4 (in Japanese). Ohtsuka, N., and Mukai, K. (1995). Proc. Int. Con$ InP and Related Materials, Hokkaido. Sakuma, Y., Ozeki, M., and Nakajima, K. (1993). J . Cryst. Growth 130, 147. Saiki, T., Mononobe, S., Ohtsu, M., Saito. N., and Kusano, J. (1996). Appl. Phys. Lett. 68,2612. Sugawara, M. (1996). Jpn. J . Appl. Phys. 35, 124. Sugawara, M., Okazaki, N., Fuji, T., and Yamazaki, S. (1993). Phys. Rev. 8. 48, 8102. Tabuchi, M., Noda, S., and Sasaki, A. (1992). Science and Technology of Mesoscopic Structure, Springer-Verlag, Tokyo, p. 379.

SEMICONDUCTORS AND SEMIMETALS, VOL 60

CHAPTER5

The Photon Bottleneck Effect in Quantum Dots Kohki Mukai und Mitsuru Suguwara OPTICAL SCMlCONDLlrTOll &VICES

LABORAT(1RY

FUJITSU LABORATORIES LTI,.

ATSUGI,KAHAGAWA. JAPAN

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. A MODELOF THE CARRIER RELAXATION PROCESS IN QUANTUM DOTS . . . . . 111. EXPERIMENTS ON LIGHTEMISSION AND CARRIER RELAXATION I N QUANTUM-DOT DISCRETE ENERGY LEVELS . . . . . . . . . . . . . . . . . . . . . . . 1. Elertroluminescence Spectra . . . . . . . . . . . . . . . . . . . . . 2. Time-Resolved Photoluminescence . . . . . . . . . . . . . . . . . . 3. Simulation of Electroluminescence Spectra . . . . . . . . . . . . . . . IV. INFLUENCEOF THERMAL TREATMENT . . . . . . . . . . . . . . . . . . . 1. Change in Emission Spectra after Annealing . . . . . . . . . . . . . . 2. Competition between Carrier Relr.rution and Recombination . . . . . . . V. SIMULATION OF LASERPERFORMANCE. INCLUDING THE AUGERRELAXATION PROCESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. SUMMARY References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I.

209 21 1 2 14 2 15 2I I 225 229 229 231 235 231 238

Introduction

The physics of carrier relaxation in quantum dots has been intensively studied ever since it was discovered that quantum dots held promise for optoelectronic device applications (Arakawa and Sakaki, 1982, 1984; Brus, 1986; Chemla and Miller, 1986; Asada, 1986). Nevertheless, many researchers feared that carrier relaxation into the quantum-dot discrete levels would be significantly slowed due to a lack of phonons to satisfy the energy conservation rule. This is the so-called phonon bottleneck problem. If we form well-separated discrete energy levels by quantum dots so as to increase laser performance, carrier relaxation into the levels may be much retarded, and therefore laser performance may not be improved. As we saw in Chapter 1, 209 Cop)righl 1 1Y99 hy AcddcmiL Preas All ri$it\ of reproductioii in m y Corm re\erved ISBN o - I z - ~ Q I ~ ~ . ~ ISSN ooxn-x7x4 99 m o o

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KOHKIMUKAIAND MITSURUSUGAWARA

the carrier relaxation lifetime dominates the performance that can be expected for quantum-dot lasers. Theoretical studies on carrier relaxation due to phonon scattering show that the relaxation lifetime of the dots reaches nanoseconds or longer unless the requirements of energy conservation are strictly satisfied (Bockelmann and Bastard, 1990). This long relaxation lifetime, which is comparable to the radiative or nonradiative carrier recombination lifetimes in semiconductors, was thought to significantly degrade radiative efficiency (Benisty, 1991; Benisty et al., 1995; Bockelmann, 1993), because carriers are consumed through radiative and nonradiative channels on their way to the ground level (Fig. 5.1). Yet there are also more optimistic theoretical results. An auger process has been suggested for a possible fast relaxation channel with a lifetime of ten fewer picoseconds (Bockelmann and Egeler, 1992; Efros et al., 1995; Gerard, 1995). Inoshita and Sakaki (1992) showed that two phonon processes, including LO and LA phonons, reduce the requirements of energy conservation. And Nakayama and Arakawa (1994) solved the time-dependent Schrodinger equation including an electron-LO-phonon interaction Hamiltonian for the relaxation problem. They noted a margin of a few tens meV for the energy conservation rule due to the time-energy uncertainty, and predicted a relaxation time of less than one picosecond.

Energy

.........................

I

I Nonradiative I

channel

I

... (Defect level)

I I I -Y.-Radiative channel

FIG.5.1. Schematic of the phonon bottleneck. Discrete sublevels in quantum dots may hinder carrier relaxation toward the ground level. In this case. carriers will recombine to emit at higher levels or go to the nonradiative channel during a carrier cascade toward the ground level.

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The recent discovery of self-assembled quantum dots composed of various semiconductor materials has enabled us to evaluate the carrier relaxation lifetime experimentally (Eaglemann and Cerullo, 1990; Tabuchi et al., 1992; Ahopelto et al., 1993; Leonard et al., 1993; Mukai et al., 1994; Oshinowo et al., 1994; Moison et al., 1994; Apetz et al., 1995; Utzmeier et al., 1996). Retarded carrier relaxation, with a lifetime of several tens to hundreds of picoseonds, has been observed in many experiments using time-resolved photoluminescence (Mukai et al., 1996b; Yu et al., 1996; Adler et al., 1996). Some of these experiments confirmed the importance of phonon resonant excitation conditions in accelerating relaxation (Raymond et al., 1995; Vollmer et al., 1996). Fast relaxation through the Auger process reportedly manifests itself in the fast rise of time-resolved photoluminescence and during the initial stage of carrier relaxation, when a large number of carriers surround the quantum dots (Ohnesorge et al., 1996, Bockelmann et al., 1997). While more detailed theoretical and experimental work is needed to reach precise conclusions about relaxation mechanisms, we are convinced that the phonon bottleneck does exist, in the sense that relaxation is slower in quantum dots than in quantum wells, where a typical intersubband relaxation lifetime is 0.1 to 1 ps (Asada, 1989). In this chapter, we present our findings on carrier dynamics in our selfassembled InGaAs/GaAs quantum dots grown by the alternate supply (ALS) process (Mukai et a]., 1994; see Chapter 3). The dot’s high crystal quality, which features narrow spectrum broadening and high emission efficiency, enabled us to pursue the problem. After a brief discussion of possible carrier relaxation processes into quantum dots, we analyze electroluminescence and time-resolved photoluminescence data to provide recombination and relaxation lifetimes. Next we introduce our experiments on annealed quantum-dot samples to demonstrate the influence of retarded carrier relaxation on emission spectra. Finally, we simulate laser performance by adding an Auger relaxation process to the rate equations in Chapter 1. The results in this chapter will clarify how the phonon bottleneck effect works in quantum dots.

11. A Model of the Carrier Relaxation Process in Quantum Dots

The carrier relaxation process into quantum dots is actually two processes, as shown in Fig. 5.2. One is the carrier relaxation from continuous energy levels into quantum-dot discrete levels (A). The other is the relaxation between the discrete levels inside dots (B). In many optical experiments as well as in quantum-dot lasers, carriers undergo these two processes

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FIG. 5.2. Two cases of carrier relaxation in a quantum dot: (A) the relaxation from continuous levels and (B) the relaxation between discrete levels. It is expected that case (A) will occur faster than case (B) since the energy conservation rule is more severe for the latter case.

except when they are brought directly into the discrete levels by resonant excitation or tunneling (Kamath et al., 1997a). Since the energy conservation rule must be satisfied for carrier relaxation, relaxing carriers transfer a corresponding energy to other particles such as phonons and to other carriers. Thus, the relaxation rate strongly depends on the density of final levels and on the number of particles other than the transition matrix elements. Figure 5.3 illustrates the two representative processes: (a) a single- or multiphonon process, and (b) an Auger process. Inoshita and Sakaki (1992) showed that two-phonon processes, including LO and LA phonons, reduce the requirements of energy conservation, thus drastically decreasing the lifetime to subpicosecond order. Even so, this relaxation process cannot send carriers deep inside the dots, since the transition matrix for the relaxation processes that include more than three phonons will be greatly reduced. Therefore, carriers once trapped in the upper levels of the dots relax into lower energy levels by emitting phonons step by step. The Auger process of Fig. 5.3(b) works effectively when there are many carriers, since an electron can find another electron or a hole into which to transfer its energy and thus fall into the dot energy levels. This process works to further relax carriers trapped in the discrete levels into lower energy regions, as long as there is still a larger number of carriers outside the dots. For that reason the Auger process is more effective for carrier relaxation into deep sublevels. Figure 5.4 shows an experimental study of the rising characteristics of photoluminescence for SK dots as a function of excitation power (Ohnesorge et al., 1996). The rising time depends on the optical excitation density. For an excitation density between lo-' and 4, rising time stays almost

5

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BOTTLENECKEFFECT IN QUANTUM DOTS

213

(a) Multiphonon process

Shallow sublevel

Deep sublevel

High carrier density

Low carrier density

FIG. 5.3. Schematic of two processes of carrier relaxation from continuous levels: (a) the multi-phonon process and (b) the Auger process. Usually, the energy separation between the continuous level and the sublevel in a dot does not exactly match the energy of a single phonon. There can be a combination of LO and LA phonons during relaxation from the continuous energy level, but this is unlikely for deep sublevels. The Auger process is effective when carrier density is high.

constant at 90ps. An increase in the excitation density to lo2 W/cm2 achieves a rising time of 40 ps. The authors claimed that the multi-phonon process is dominant while the rising time is constant; the decrease of rising time is due to the Auger process. In time-resolved photoluminescence, a large number of carriers, many more than the number of dots, are excited by a short intense laser pulse. Then, within a t least several tens of picoseconds, most carriers are captured to occupy the quantum-dot levels, primarily via the Auger process. This is followed by a carrier relaxation process in individual dots toward lower energy levels via phonon emission. In photoluminescence and electroluminescence, where there are fewer carriers than quantum dots, carrier injection and relaxation into dots is dominated by phonon emission. As a result, carriers are first captured by upper energy levels and then relax as if they were going down a ladder of discrete energy levels. The capture process occurs at a relatively high rate, since the transition is from continuous to discrete levels and the energy conservation requirements are easily satisfied by the multi-phonon process. However, the intersubband relaxation is

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120

4 I

100

T=5K

I

e 80

..

0

.-

a

40

0

2oL

I

lo-’

'

’’mns-s’

100

'

' ' 1 4 1 1 6 1

'

' ' s m m a m '

10'

8

1

102

Excitation density (W/cm') FIG.5.4. Experimental study of rising characteristics by Ohnesorge et al. (1996). The authors showed that the rising time depended on the excitation density in the SK dots. They reported that about 4 W/cmZwas the critical excitation density for the Auger carrier relaxation process (from Ohnesorge et al., 1996. Copyright 1996 by the American Physical Society).

significantly slowed as long as the energy separation closely matches the phonon energies. During laser operation, at a low injection level even above the threshold, carrier relaxation relies on the ladder process. As the injection level increases, the number of carriers around the dots increases, making the Auger process effective for carrier relaxation.

111. Experiments on Light Emission and Carrier Relaxation in Quantum-Dot Discrete Energy Levels An experimental clue to the phonon bottleneck problem was given in the electroluminescence spectra of Fig. 4.7 in Chapter 4, where well-resolved multiple-peak light emissions from discrete energy levels were observed in the ALS dots. We increased the current injection and found that well before the emission from the ground level is saturated, a higher energy emission occurs that exceeds the ground-level emission. The carrier distribution between the discrete levels in no way follows the Fermi-Dirac distribution for the measured temperature of 77 K. The most probable explanation for this is that the carrier relaxation lifetime is comparable to the carrier recombination lifetime.

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BOTTLENECKEFFECTIN QUANTUM DOTS

215

After discussing the electroluminescence spectra in a little more detail, we will evaluate the carrier relaxation and recombination lifetimes in ALS quantum dots by time-resolved photoluminescence. The emission decay curve from the excited levels is found to be double exponential, while the decay from the ground level follows a single exponential curve. The decay curves are analyzed by a model that takes into account a rapid initial carrier capture process and the independence of each quantum dot. This model enables us to obtain the relaxation lifetime as well as the recombination lifetime. The unique electroluminescence spectra are well simulated by the determined lifetimes.

1. ELECTROLUMINESCENCE SPECTRA Figure 5.5 shows the electroluminescence spectra for (a) 300 K and (b) 77 K. To fabricate diode structure, a 0.5-pm n-GaAs layer, a 1-pm n-InGaP layer, an ALS-dot layer sandwiched by 100-nm GaAs layers, a l-pm p-AlGaAs layer, and a 0.5-pm p-GaAs layer were grown on a (001) GaAs substrate. The luminescence parallel to the sample surface was dispersed and detected with a lock-in technique using an InGaAs photo-multi detector that was kept at -70°C. The electrode size was 20 x 900pm’. The currentinjected area was estimated by the near field pattern, and according to this estimation, it was spread out over 150 to 200 x 900pm2 in the quantum-dot layer. Three emission peaks corresponding to the interband transition between the discrete levels of the conduction and valence bands appeared at 300 K when the injected current was increased, five peaks appeared at 77 K. In Chapter 4, we pointed out the phonon bottleneck from the emission spectra at 77 K. We should note that, at lOmA, the 2nd and 3rd peak emissions exceeded the first peak emission much earlier than the first emission intensity saturated, indicating that the relaxation rate into the ground level is comparable to the spontaneous emission rate of the excited levels. The electroluminescence spectra of Fig. 5.5 varied between the two temperatures, which indicates that some features of carrier dynamics depend on temperature. For example, when the emission intensity of the ground level reached half maximum, the intensity of the second level was stronger than that of the ground level at 77 K (10 mA), but the two were almost equal at 300 K (50 m A). Random carrier capturing into the dots was proposed by Grundmann and Bimberg (1997) as an alternative interpretation for the multiple emission peaks (Mukai et al., 1996b; Lipsanen et al., 1995). They calculated the emission spectra using a Poisson distribution to count the number of

216

KOHKIMUKAIAND MITSURU SUGAWARA

0.8

0.9

1.0

1.1

1.2

1.3

Energy (eV)

(a> l""l""l""l""

- Measurement

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 * 1 1 1

0.9

1.0

1.1

1.2

1.3

1.4

Energy (eV)

(b) FIG. 5.5. Electroluminescence spectra at (a) 300 K and (b) 77 K. As the injected current was increased, five discrete levels appeared. Higher-level emissions appeared before the emission intensity of the lowest level reached its maximum value. Considering that the kT value is much narrower than the broadening of the spectra, these figures suggest the existence of a phonon bottleneck that impedes carrier relaxation (from Mukai et al., 1996b. Copyright 1996 by the American Physical Society).

5 PHOTON BOTTLENECK EFFECTI N QUANTUM DOTS

217

carriers that were trapped in quantum dots. As a result, they claimed that, even if the retarded carrier relaxation is not taken into account, the emission from high energy levels appears even before the lower energy level emission is saturated. However, their calculation of the emission spectra cannot explain the emission spectra we measured, where the third-level emission exceeded the ground-level emission before the ground-level intensity saturated. Above all, the random capture model offers no explanations for the difference in the emission spectra between 77 K and 300 K.

2. TIME-RESOLVED PHOTOLUMINESCENCE

To evaluate carrier relaxation and recombination dynamics, we measured the time-resolved photoluminescence of ALS quantum dots. The dots were grown by 18 cycles of In-Ga-As alternate supply onto a 500-nm GaAs buffer layer on a (001) GaAs substrate and capped b y a 100-nm GaAs layer. The growth temperature was 460°C. Samples set in a temperature-controlled cryostat were excited by a mode-locked 532-nm Nd:YAG laser beam with a pulse width of 200 ps and a repetition rate of 82 MHz. Luminescence from the sample surface was dispersed by a monochrometer and time-resolved with a streak camera. Figure 5.6 shows typical experimental results of luminescence intensity decay for up to the fifth emission peak at 20 and 250K. The order of emission peaks corresponds to that in the electroluminescence spectra of Fig. 5.5(b). Various features can be noted in the decay curves. First, the luminescence decays faster as the order of emission peaks gets higher. Similar results were obtained by Raymond et al. (1996). Since the spontaneous emission lifetime due to the interband transition in quantum dots is common for each energy level if we assume a constant oscillator strength (Chapter l), the observed variation of the decay curves suggests that the carrier relaxation lifetime decreases with increasing order. Second, the emission decay is nonexponential, which is probably due to the fact that the decay is a combination of relaxation and recombination components. Third, the temporal evolution of the decay is very fast during the initial stage, but becomes eventually very slow at the end (Mukai et al., 1996b; Adler et al., 1996; Yu et al., 1996). This is in contradiction to the calculated curves, which are moderate at the beginning but become fast toward the end (Grundmann and Bimberg, 1997; Grosse et al., 1997). Fourth, the decay is faster at 250 K than at 20 K, which indicates that relaxation and/or recombination lifetime is shorter at higher temperatures. To analyze the decay curves, we introduce a new model that we call random initial occupation (RIO), which is appropriate for analyzing time-

218

KOHKIMUKAIAND MITSURUSUGAWARA

I " " " " " " " ' 1

0

2

4

6

8

Time (ns) FIG. 5.6. Example of luminescence decay of the five levels at 250 and 20K. The higher the temperature, the faster the decay; the lower a level, the slower the decay (from Mukai et al., 1996b. Copyright 1996 by the American Physical Society).

resolved photoluminescence data and is based on the following two points. First, assuming that all carriers relax into the quantum-dot discrete levels during pulse excitation, we start the calculation of the decay curve at the point where levels inside the dots are occupied by carriers. This is reasonable because there are a large number of carriers within 10" to 1013cm-2 around the dots during the short pulse excitation, and carrier-carrier scattering causes rapid relaxation into dots through the Auger process. Second, we take into account that the carrier capture process occurs randomly and that the number of carriers and their distribution in discrete levels differs from dot to dot. Taking into account only the two electron levels of ground and excited for reasons of simplicity, the initial carrier distribution in dots can be categorized into the four types shown in Fig. 5.7(a): A, both levels are filled; B, only the excited level is filled; C , only the ground level is filled; and D, both levels are empty. The temporal evolution of this system is described in the figure. Case D is the final situation of every carrier process. When a dot A,, and A,-are possible for is initially in case A, three paths-A,,

5

PHOTON

A

BOTTLENECK EFFECTIN QUANTUM DOTS

[E]I?[

B

C

D

a

D

I

I

I

I

I

I

I

I

t l

I

I

I

I

I

1.0

2.0

3.0

4.0

I

I

219

j

3

0

> 0

NBO

3

c

NAO

.3

k E 1

D

c

0

(b)

5.0

Time (ns)

FIG. 5.7. (a) Model for the carrier dynamics between two levels in an uncoupled dot. When the carrier injection from continuous levels into sublevels is sufficiently fast, all possible initial situations can be classified as four cases, A, B, C, or D. The decay in the carrier number in the excited level is described by considering paths A , , A,, A,, B,, and B,. ( b ) Calculated number of carriers in the excited levels as a function of time. The solid line shows N(r), the dashed lines show N,(t) and N,(t). The figure shows a double exponential decay line, which matches the measured results (Mukai et al., 1996; Adler et al., 1996). The faster decay indicates a carrier process that begins at case B, and the slower decay indicates one that begins at case A (from Mukai et al., 1998).

220

KOHKIMUKAIAND MITSURUSUGAWARA

reaching the final destination of D. Moreover, two paths, B, and B,, are possible for case B, and a single path, C,, is possible for case C. Radiative decay is allowed in both levels on the assumption that hole quantization energies are less than k,T due to the comparative heavy mass of holes (Benisty et al., 1991). The carrier number in the excited level is counted by evaluating the paths beginning from case A and the paths beginning from case B. Let NAA(t)be the number of A at time t; and NAB@),the number of B formed from A as a result of the recombination of the ground-level carrier. The carrier number in the excited level in dots when the initial situation is A is described by

Here, the rate equation for N A A ( t )is

where z, is the carrier recombination lifetime, which is assumed to be common between the two levels, Considering both paths A , and A,, the rate equation for NAB(t)is given by

where zreLis the carrier relaxation lifetime from the excited to the ground level. Solving Eqs. (5.2) and (5.3), we obtain

(5.4)

where NAo is the initial number of case-A dots. When zrel<< z,, NAB(t) becomes zero and the decay lifetime of N A ( t ) reduces to 2 , / 2 . When zrel is not negligible compared with z, the decay of NAB(t)is not negligible and the decay lifetime of NA(t) is larger than r r / 2 . For the carrier number in the excited level when the initial situation is B,

5

PHOTON

BOTTLENECK EFFECTIN QUANTUM DOTS

221

NB(t), the rate equation is

and we obtain

where N,, is the initial number of case B. The decay lifetime of NB(t) is ~ ~ - ‘ ) -and l gets closer to sret when z~, << T ~ .As a result, the total number of carriers in the excited level is given by

(T~;:

+

Ne.x(t) = N A ( t )

+ NB(t)

(5.8)

The emission intensity is given by Ne,(t)/z,. The measured double-exponential decay curves in Fig. 5.6 are reproducible by this type of calculation. The solid curve in Fig. 5.7(b) shows the decay in the number of carriers in the excited levels as calculated by Eq. (5.8) with T~ = 3 ns, t r e= l 200 ps, and NAo = N,,/8 as parameters. The resultant solid curve consists of two dashed lines: steep, reflecting the decay starting from case B (NB(t));and moderate, reflecting the decay starting from case A (NA(t)).The critical point when the inclination of the line changes depends on the ratio of NAo/NB,,.For N,, > NAo, we get a decay curve with fast and slow components. The carrier number in the ground level is also calculated by considering the paths A,, A,, B,, and C,. As for the excited level, let us define the carrier numbers NABC(t),NAC(t),NBc(t),and N,(t) as shown in Figs. 5.7(a). The rate equations for these are

(5.10) (5.11)

and

(5.12)

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KOHKIMUKAI AND MITSURUSUGAWARA

Solving Eqs. (5.9) through (5.12), we obtain (5.13) (5.14) (5.15) f

Nc(r) = Nc,ept

(5.16)

where N,, is the initial number of carriers in case C. Finally, the total number of carriers in the ground level is given by

<< t,, Eqs. (5.13) through (5.15) become zero and the decay time When treI of Ng,(t) reduces to t,. In the same case, we saw that the decay lifetime of NA(t) reduces to 2,/2. However, in Fig. 5.6, the later moderate decay in the second level is not half of the decay in the ground level. This indicates that the relaxation lifetime is not negligible compared with the recombination lifetime and/or that the recombination lifetime may not be strictly uniform among the sublevels. We analyzed the measured decay curves of Fig. 5.6 by the doubleexponential formulation of

(5.18) after deconvoluting the decay curves from the excitation pulse. The second term of Eq. (5.18) is of course neglected in considering the decay of the ground level. Here, from the estimation by the RIO model we assumed the initial steep decay component to be dominated by the carrier relaxation and the subsequent slow decay component to be by the recombination. In strict treatments, the degeneracy and the selection rule in relaxation for the multiple sublevels must be taken into account in solving the rate equations. However, Eq. (5.18) is a good approximation because the initial situation of the dots is roughly categorized into two cases: when the lower level is empty and when the lower level is filled.

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PHOTONBOTTLENECK EFFECTIN QUANTUM DOTS

223

The results of the analysis are plotted in Fig. 5.8 as a function of temperature. The recombination lifetime in Fig. 5.8(a) is independent of level order and temperature. The relaxation lifetime of Fig. 5.8(b) varies among the levels and is temperature-dependent. The nano-second recombination lifetimes and their temperature independence match a theory for the spontaneous emission of electron-hole pairs in quantum dots that was discussed in Chapter 1. Similar experimental results were obtained for

. Y 3

4 ;10 0 .C Y

m

c ._

0

50

100 150 200 250

300

Temperature (K)

(4

10p-

0

SO 100 150 200 250 300

Temperature (K)

(b) Re 8

,

FIG. 5.8. Carrier lifetimes of sublevels as a function of temperature: (a) recombination lifetime and (b) relaxation lifetime. The recombination lifetime is independent of temperature and about 1 ns in all sublevels. The higher the temperature and the higher the order of a level, the faster the carrier relaxation (from Mukai et al., 1996b. Copyright 1996 by the American Physical Society).

224

KOHKIMUKAIAND MITSURUSUGAWARA

InGaAs/GaAs SK dots by Wang et al. (1994) and Kamath et al. (1997b) and for InP/InGaP SK dots by Kurtenbach et al. (1995). As the order of levels rose from the second to the fifth, the relaxation lifetime decreased by one order of magnitude. This is probably because the density of levels increases with the level order, as seen in the electroluminescence spectra, which increases the probability that carriers will relax into the lower empty levels. The decrease in relaxation lifetimes following a temperature decrease can be attributed to the increase in the number of phonons following the Bose distribution function. Let us check what the decay curves look like if we assume that carriers relax from the continuous level to the topmost fifth level and then relax step by step. The rate equations for this case are (5.19)

(5.20)

and (5.22) where T,, is the recombination lifetime in the continuous level, z, is the relaxation lifetime from the continuous level to the topmost fifth level; and 7i denotes the relaxation lifetime from the ith sublevel to the (i - 1)th sublevel. Using the occupation factor of fi = Ni/Di with the level density of D i , we get

where zi0 is the intrinsic carrier relaxation lifetime at the ith level. Using the lifetimes in Fig. 5.8, we calculated the temporal evolution of the carrier number in the sublevels (Fig. 5.9). To estimate the fastest rise under step-by-step relaxation, we assumed ,h = 0 in the calculation and found that the delay of rising at the lower-ordered level cannot be neglected. The delay is about 2 ns between the fifth and the ground level, and is approximately equal to the simple sum of the relaxation lifetimes of the sublevels.

5 PHOTON BOTTLENECK EFFECT IN QUANTUM DOTS

I"

0

0.5

1 Time (ns)

1.5

225

2

FIG. 5.9. Calculated rise of the carrier number when the direct carrier relaxation from the continuous level is negligible. The rise slowed as the level order decreased. The delay is approximately equal to the simple sum of the relaxation lifetimes of the sublevels.

Comparing the experimental results of Fig. 5.6 and the calculated results of Fig. 5.9, it is clear that the assumption of the step-by-step relaxation process cannot explain the fast rising of photoluminescence in the time-resolved measurements. The RIO model, which assumes fast carrier setup in quantum dots resulting from carrier-carrier scattering and the subsequent slow decay inside the dots, successfully explains the decay curves of time-resolved photoluminescence.

3.

SIMULATION OF

ELECTROLUMINESCENCE SPECTRA

Let us simulate the electroluminescence spectra of Fig. 5.5 using the evaluated relaxation and recombination lifetimes. Note that the carrier number around the quantum dots is estimated to be about two to three orders of magnitude smaller than that in time-resolved photoluminescence measurements. Thus, we employ the step-by-step relaxation model, where the carriers first relax into the topmost level and then relax into the low energy levels. We ignore the thermal excitation of electrons, since the level spacing is much larger than the value of kBT The rate equation for the carrier number in sublevels during electroluminescence measurement is obtained from Eqs. (5.20) through (5.22) by

226

KOHKIMUKAIAND MITSURUSUGAWARA

assuming that the differential of time is zero, N5

-

N5 - G = O , +-

55

Ni N . N , , , -+>--- 0 (i Ti

(5.24)

‘r

zr

Ti+

= 2,

3,4),

(5.25)

1

and (5.26) where G is the carrier injection rate from the continuous level into dots. Here, we consider that the recombination lifetime, zr, is common to all levels, as seen in Fig. 5.8(a), and disregard nonradiative recombination in our high-quality samples (Chapter 4).Assuming a Gaussian distribution for the broadening of the emission spectrum, we calculated the luminescence spectra by

(5.27) where E is the energy of the photon, E , is the energy at the ith level, and r is an inhomogeneous broadening factor. Figures 5.10(a) and 5.10(b) show the results of the calculation for 300K and 77K, respectively. During calculation, we used an in-plane dot density of 5%, a broadening factor of 50meV, T~ = 1 ns, and D i= 2 . i ( i = 1 to 5), taking the spin into consideration. The intrinsic relaxation lifetimes used were zo2 = 400, zo3 = 200, T~~ = 50, and z05 = 20ps for 300K; and T~~ = 1000, T~~ = 500, T~~ = 200, and z 0 5 = 90 ps for 77 K. The calculations of Fig. 5.10 explain the measured electroluminescence spectra of Fig. 5.5. The relaxation lifetime is short enough that the groundlevel emission is dominant for small currents. As the number of injected carriers increases, the carrier relaxation rate decreases and emission from higher levels occurs. Since the relaxation lifetime is closer to the radiative recombination lifetime at 77K than at 300K, the emission from higher levels at 77 K occurs for a smaller current. At 77 K, the calculated current values match the measured values. At 300K, the trend of relative peak intensities is well explained, but the values for the current by the calculation are much smaller than the measured values. This discrepancy grows larger as the current increases- a tendency that can be attributed to the consump-

5

PHOTON

BOTTLENECK EFFECTIN QUANTUM DOTS

0.9

I0

1.1

1.2

1.3

221

1.4

Energy (eV)

(b) FIG. 5.10. Calculated electroluminescence spectra at (a) 300 and (b) 77 K using the rate equations of the five levels, The results of the calculation correlate with the measured spectra in 5(a) and (b) (from Mukai et al., 1996b. Copyright 1996 by the American Physical Society).

tion of carriers due to nonradiative recombination via defects outside the dots (see Chapter 4), and to a rise in local temperature due to the nonradiative process -leading to further acceleration of nonradiative recombination. Figure 5.11 shows the values of f;. as a function of the current for (a) 300K and (b) 77K, respectively. Note that the ground, second, and third

228

KOHKIMUKAIAND MITSURUSUGAWARA

(a>

(b)

0

50

100 150 200 250 300

Current (mA)

Current (mA)

FIG. 5.1 1. Value of f; as a function of injected current for (a) 300 K and (b) 77 K. The ground, second, and third levels were not completely filled, and the maximum occupation factor of these levels depended on the temperature.

levels are not completely filled and that the characteristic features depend on temperature. For example, the maximums of f i for the ground and the second levels are about 80% at 300 K and about 60% at 77 K. For the third level, these maximum are 90% at 300 K and 80% at 77 K. While the emission from the ground level is dominant, the maximum occupation factor at the ground level is 60% for 300 K. In quantum-dot lasers, the suppression of carrier filling may limit gains needed for lasing at the ground level (Shoji et al., 1995).

5

PHOTON BOTTLENECK EFFECTIN QUANTUM DOTS

229

IV. Influence of Thermal Treatment The electroluminescence spectra with a series of emission peaks are dominated by the balance between carrier relaxation and recombination rates, as explained in the previous section. Here, we introduce an interesting example of this effect found in annealed quantum dots. 1. CHANGE IN EMISSION SPECTRA AFTER ANNEALING

Annealing of ALS quantum-dot samples was performed in a reactor tube of a liquid-phase epitaxy system between 500 and 68OoC, in a pure N, gas atmosphere so as to protect the samples from oxidation. The quantum dots were originally grown at 460°C using the growth sequence of 18 cycles of In-Ga-As (see Chapter 3). A GaAs plate was placed on a sample surface to raise the arsenic vapor pressure and thus prevent arsenic desorption from the sample surface. Figure 5.12 compares the photoluminescence spectrum of an as-grown sample with the spectra of samples after annealing at 615°C for 30 and 60

""'""'"""""""'1

h .r( Y

2

Y

E

H

1.o

1.1

1.2

1.3

1.4

1.5

Wavelength (pm) FIG. 5.12. Photoluminescence spectra before and after annealing at 615°C for 30 and 60 minutes. The emission from the sublevels is observed. The integrated photoluminescence intensity decreased after annealing. It is remarkable that the emission intensity of the ground level decreased several times over that of the second level (from Mukai et a!., 1998).

230

KOHKIMUKAIAND MITSURUSUGAWARA

minutes. In the as-grown sample, three peaks can be observed that are due to the interband transition between the discrete levels of the conduction and valence bands. The excitation power is very low, as the emission intensity of the ground level in Fig. 5.12 is less than one-fifth of the maximum value reached at a higher excitation. Annealing greatly reduces the integrated photoluminescence intensity, broadens the spectra, and slightly moves the emission peak toward higher energies. What is most remarkable is that the relative peak intensity between the ground- and second-level emission is greatly changed after annealing: While the ground-level emission is about three times stronger than the second-level emission in the as-grown sample, the two peaks are almost comparable in their intensity after 30-minute annealing. This is not due to drastic structural changes and the resultant reduction of the transition oscillator strength, since the emission energy shift is only about 5 meV and the TEM image of the 30-minute-annealed samples shows no significant changes in dot structure, size, and numerical density. One possible explanation for this phenomenon is that the carrier dynamic process in quantum dots is influenced by the thermal treatment, which leads to the change in the relative intensity. Figure 5.13 shows the energy of the ground- and second-level emission peaks, as well as the relative emission intensity of the ground level against

Annealing temperature (iC) FIG. 5.13. Emission intensity of the ground level divided by that of the second level (Iground/12nd) and emission energy of the ground and second levels as a function of annealing temperature. At over 600"C, the energy shift occurred, and the relative intensity of the ground level was greatly reduced after annealing. The energy shift was larger in the second level up to 650°C (from Mukai et al., 1998).

5

PHOTONBOTTLENECK EFFECT IN QUANTUM DOTS

231

the second level, as a function of the annealing temperature. The energy shift occurs at over 600°C because of interdiffusion, and simultaneously the relative intensity of the ground-level emission is greatly reduced. The figure also shows that the energy shift is larger in the second peak and that the level separation is increased by annealing. This contradicts our experiences in the study of interdiffusion in quantum wells, where the level separation always decreases after interdiffusion, which spreads the confinement potential in the growth direction. This unique feature can be attributed to structural asymmetry between the in-plane and perpendicular directions in ALS quantum dots with a disk-like shape because of the asymmetry, the interdiffusion in the early stage occurs faster at the edge of the dot than in the central region (Sallese et al., 1995). Since this corresponds virtually to the decrease in the in-plane diameter of the dots, the energy separation increases.

2.

COMPETITION BETWEEN CARRIER

AND

RELAXATION

RECOMBINATION

Figure 5.14 shows examples of time-resolved photoluminescence of the ground- and second-level emission after annealing at 615°C for 60 minutes. The decay time of the ground-level emission decreases as the temperature rises. This is in contrast to the decay times of as-grown samples, which are insensitive to temperature (Fig. 5.8(a)). We see that the decay curves of the second level are double-exponential, a result of the competition between the relaxation into the ground level and the recombination at the second level, as seen in Section 111. As the temperature increases, the relaxation accelerates, but the recombination lifetime is insensitive to temperature. Figure 5.15 shows (a) recombination and (b) relaxation lifetimes up to the third level before and after annealing. The solid and open symbols indicate the lifetimes of as-grown and annealed samples, respectively. The third level in the annealed sample is not observed clearly because of emission broadening. The recombination lifetimes at the ground level become temperature -dependent after annealing and decrease to a few hundred picoseconds near room temperature. This suggests that the nonradiative channel affects only the ground level. The nonradiative lifetime for the ground level is estimated to be 250 ps if we use the recombination lifetime of 200 ps and the radiative lifetime of 1 ns. The relaxation lifetimes decrease to about half, and it is not clear whether this decrease is significant since, following the RIO model we used in the analysis, the faster decay time is not a pure carrier relaxation lifetime. However, we should note the possibility that the nonradiative levels assist carrier relaxation, which was predicted by Sercel (1995).

232

KOHKIMUKAl

6

' I

I

" I " '

' 1 '

AND I '

MITSURUSUGAWARA ' I "

'

"

"

I '

I '

'

' 1

' ' I "

'g

h

3

'a1 d

2

W

5

.H

Ei

B E

Y

0

2

4

6

8

10

12

14

16

18

20

Time (ns) FIG.5.14. Examples of decay curves of the ground and second levels after annealing at 615°C for 60 minutes. The decay time of the ground level decreased as the temperature rose, but was insensitive to temperature in as-grown samples. The decay curves of the second level were double-exponential(from Mukai el al.. 1998).

We consider several possibilities to explain why the nonradiative channel affects only the ground level. One is that carrier trapping may require some phonon scattering and the probability of scattering is low for the higher levels. Another is that defect levels outside the dots, resonant in energy with the ground level, are the origin of the nonradiative channel (such resonant carrier trapping was reported by Tsang (1978)). The photoluminescence spectra can be simulated by the same procedure as in Section 111. The rate equation for the ground-level population should be changed to include the nonradiative process as (5.28)

where znlis the nonradiative lifetime at the ground level. Figure 5.16 shows the photoluminescence spectra as calculated using the nonradiative lifetime as a parameter. We used a carrier injection rate of G = s - l per individual dot, a broadening factor with a Gaussian distribution of 50 meV, z, = 1 ns, Di = 2 . i (i = 1,2,. . . , 5 ) , T~~ = 200, zo3 = 100, T~~ = 50, and zo5 =

5 PHOTON BOTTLENECK EFFECTIN QUANTUM DOTS Gv

. . , . .. . , . . .

10-8

.

,

.

,

.

.

,

.

I . ,

I

0

't 0

233

j

.., . i

300

Temperature (K)

-

A

o

so

100

is0

200

Temperature (K)

250

:t 300

FIG. 5.15. Lifetimes up to the third level as a function of temperature: (a) recombination lifetimes and (b) relaxation lifetimes. The solid and open symbols indicate lifetimes of as-grown and annealed samples, respectively. The recombination lifetimes in the ground level became temperature-dependent after annealing and decreased to a few hundred picoseconds near room temperature. The relaxation lifetimes decreased slightly after annealing (from Mukai et al., 1998).

20 ps. Unknown relaxation lifetimes of T~~ were assumed to be half of that used in Section 111, since the present experiments show that the relaxation lifetimes of T~~ decrease by about half after annealing. Unknown relaxation lifetimes of T~~ and zO5 do not affect the results of Fig. 5.16 as long as the emission from the fourth and fifth levels does not appear. The injection rate was chosen to give a low injection condition, where the fourth and fifth excited levels do not apper in the calculated spectrum. The

234

KOHKIMUKAIAND MITSURUSUCAWARA

1 .o

1.1

1.2

1.3

1.4

1.5

Wavelength (pm) FIG. 5.16. Photoluminescence spectra calculated by assuming values for the nonradiative lifetime (tnl). As the nonradiative lifetime decreased from infinity to a few hundred picoseconds, the emission intensity of the ground level decreased and became comparable to that of the second level (from Mukai et al.. 1998).

occupation at the ground level, fi, is about 8%. As the nonradiative lifetime decreases from infinity to a few hundred picoseconds, the emission intensity of the ground level decreases and becomes comparable to that of the second level. The value of the nonradiative lifetime in the simulation matches the value estimated in Fig. 5.15, using the recombination lifetime of 200 ps and the radiative lifetime of 1 ns. The carrier dynamics in the ALS dots are summarized in Fig. 5.17. The relative emission intensity of the ground and second levels is determined by the equilibrium between the radiative recombination process (rl, r2), the relaxation process (rel), and the nonradiative recombination process (nl, n2). In an as-grown sample, the nonradiative lifetimes are negligible, which can be expected from the temperature-independent radiative lifetimes (see Section 111). After annealing, the recombination lifetime at the ground level decreases due to the nonradiative channel, while the nonradiative process of the second level is still negligible. The drastic change in relative emission intensity demonstrates that the balance of radiative and nonradiative recombination and relaxation rates determines the shape of the emission spectrum. This contradicts our experience with quantum wells, whose emission spectra generally become narrow toward a lower energy

5

PHOTON

BOTTLENECK EFFECT IN QUANTUM DOTS

235

Energy

t

I I

I I

r2

channel

channel

>Radiative channel

FIG. 5.17. Schematic of carrier dynamics in a quantum dot before and after annealing. The relative emission intensity of the ground and second levels was determined by the equilibrium between the radiative recombination process ( r l . r2), the relaxation process ( r e / ) , and the nonrddiative recombination process (rrl, n2). The nonradiative lifetimes, T,, and rn2, were negligible for the as-grown sample. After annealing, the nonradiative lifetime T , , ~ .became effective. When carriers were not sufficiently supplied to the ground level, the emission intensity of the ground level decreased compared to that of the second level.

regime as the emission intensity decreases. In quantum wells, an intersubband carrier relaxation is much faster than the recombination process.

V. Simulation of Laser Performance, Including the Auger Relaxation Process From what we have seen in the experiments and their respective analyses, we can construct a framework of the carrier relaxation problem as follows. First, when there are a large number of carriers of 10'2-10'3 cm-2 around quantum dots, as in the initial stage of time-resolved photoluminescence, the carrier capture into dots occurs with a considerably short lifetime. Also, carriers are relaxed into deep low-order levels through the Auger processes. When there are only a few carriers of less than 10" cmP2, they are first trapped at the high energy levels and relax toward the lower energy levels later. During quantum-dot laser operation, the actual relaxation process

KOHKIMUKAIAND MITSURUSUGAWARA

236

varies between the above mentioned two cases as the current injection increases. Let us here introduce the Auger effect into the simulation presented in Section V.l of Chapter 1. The same rate equations are used, and we take into account the dependence of the carrier injection time into dots, zo, on the carrier density in the quantum well Nq as

where C is the Auger coefficient. Figure 5.18(a) shows the output power as a function of the injected current for C = 1.0 x cm4/s and C = 0 with zC0= 100 ps. The inhomogeneous broadening is To = 20 meV, and the coverage is = 10%. The recombination lifetimes we used are zr = zSpon= 2.7 ns and z~~= 3 ns (solid lines), and 7 r = 0.5 ns and z~~ = 0.5 ns (dashed lines). It is clear that the Auger process increases efficiency, lowers threshold current, and enables higher output power (again, calculation stops when the quantum-well carrier density reaches 1.0 x 10l2cm2). Figure 5.18(b) shows the quantum-well carrier density as a function of injected current. For C = cm4/s, zo is superimposed. We see that z0 decreases close to 10ps as the injected current increases and that the increase in carrier density is

--

(a)

Injected current (mA)

-.

(b)

l r =0.5

ns, T

~

0.5 ~ ns =

Injected current (mA)

-- .-

FIG.5.18. (a) Output power as a function of the injected current calculated with and ... . .me . rr . . * .. witnout Auger enecr. I ne spectrum oroaaening was ro= LU mev, ana me coverage was 5 = lo%, which gave P = 0.84 at the lasing threshold (from Sugawara, 1998). (b) Carrier density in a quantum well located around the quantum dots as a function of injected current. The carrier relaxation lifetime into dots decreased close to l o p s as the injected current increased when C = 1 x lo-'' cm"/s (from Sugawara, 1998).

.

.

I

.

.

5

PHOTON

BOTTLENECK EFFECT IN QUANTUM DOTS

,

231

, 1 1 1 1 1

10-

Auger coefficient (cm4/s) FIG. 5.19. Maximum output power as a function of the Auger coefficient for various quantum-dot coverages at Po = 20 meV (from Sugawara, 1998).

considerably restricted. Figure 5.19 shows the maximum output power as a function of the Auger coefficient. For a power of 30 mW, we need C = 2.1 x lo-'" cm"/s at 5 = 40% and C = 6.0 x lo-'" cm"/s for 5 = 10%. Uskov et al. (1998) calculated the Auger coefficients as a function of the quantum-dot diameter and showed that the C value reached 10-12cm4/s for both the electron and the hole capture processes via electron scattering. Whle this result looks quite promising with respect to static performance values such as external quantum efficiency and maximum power, further acceleration is required for high-speed modulation (see Chapter 1). To check the Auger coefficient values suggested by these theoretical calculations, and to design quantum-dot active layers to enable fast relaxations, we need experimental studies that concentrate on carrier relaxation dynamics under various carrier densities around quantum dots. The relaxation lifetime that we achieve determines the performance of quantum-dot lasers and the application fields in which they might be used.

VI. Summary

We presented our experimental studies on carrier dynamics in selfassembled quantum dots and discussed their influence on lasing performance. Electroluminescence and time-resolved photoluminescence data were

238

KOHKIMUKAIAND MITSURUSUGAWARA

analyzed to provide recombination and relaxation lifetimes as a function of temperature. The RIO model in the analysis explained the double-exponential decay of excited levels, and, using the obtained lifetimes, electroluminescence and photoluminescence spectra were successfully simulated. One thing we should stress is that the phonon bottleneck problem does exist, as we experimentally found that carrier relaxation between discrete energy levels was very slow with lifetime above 100 ps. Also, carrier-carrier scattering can send carriers deep inside the dots, as seen in the initial stage of time-resolved photoluminescence decay. These two findings, we believe, help to clear up the confusion in this research field. As shown in the laser simulation, high static performance of quantum-dot lasers can be expected if the Auger process accelerates the relaxation up to about 10 ps.

REFERENCES Adler, F., Geiger, M., Bauknecht, A., Scholz, F., Schweizer, H., Pilkuhn, H. M., Ohnesorge, B., and Forchel, A. (1996). J. A p p l . Phys. 80, 4019. Anopelto, J., Yamaguchi, A. A,, Nishi, K., Usui, A., and Sakaki, H. (1993). Jpn. J . Appl. Phys. 32, L32. Apetz, R.,Vescan, L., Hartmann, A,, Dieker, C., and Luth, H. (1995).A p p l . Phys. Lett. 66,445. Arakawa, Y., and Sakaki, H. (1982). Appl. Phys. Lett. 40, 939. Arakawa, Y., Vahala, K., and Yariv, A. (1994). Appl. Phys. Lett. 45, 950. Asada, M., Miyamoto, Y.,and Suematsu, ?. (1986). IEEE J . Quantum Electron. QE-22, 1915. Asada, M. (1989). IEEE J . Quantum Electron. 25, 2019. Benisty, H., Sotomayor-Torres, M. C., and Weisbuch, C. (1991). Phys. Rev. B. 44, 10945. Benisty, H. (1995). Phys. Rev. B. 51, 13281. Bockelmann, U., and Bastard, G. (1990). Phys. Rev. B. 42, 8947. Bockelmann, U., and Egeler, T. (1992). Phys. Rev. B. 46, 15574. Bockelmann, U. (1993). Phys. Rev. B. 48, 17637. Bockelmann, U., Heller, W., Filoramo, A,, and Roussignol, P. (1997). Phys. Rev. B. 55, 4456. Brus, L. (1986). IEEE J. Quantum Electron. 22, 1909. Chemla, D. S., and Miller, B. A. D. (1986). O p t . Lett. 11, 522. Eaglemann, D. J., and Cerullo, M. (1990). Phys. Reti. Lett. 64, 1943. Efros, A. L., Khachenko, A. V., and Rosen, M. (1995). Solid Stute Cornmun. 93, 281. Gerard, J.-M. (1 995). Con$ned Electrons and Photons: New Physics und Applicutions. Plenum, New York, p. 357. Grundmann, M., and Bimberg, D. (1997). Phys. Rer. B. 55,9740. Grosse, S., Sandmann, H. H. J., Plessen, V. G., Feldmann, J., Lipsanen, H., Sopanen, M., Tulkki, J., and Ahopelto, J. (1997). Phys. Rev. B. 55, 4473. Inoshita, T., and Sakaki, H. (1992). Phys. Rev. B. 46, 7260. Kamath, K., Klotzkin, D., and Bhattacharya. P. (1997a). Proc. Confi IEEE Lasers and Electro-Optics Society, p. 498. Kamath, K., Chervela, N., Linder, K. K., Sosnowski, T., Jiang, H.-T., Norris, T., Singh, J., and Battacharya, P. (1997b). Appl. Phys. Len. 71, 927. Kurtenbach, A., Ruhle, W. W., and Eberl. K. (1995). Solid State Corn. 96, 265.

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BOTTLENECK EFFECTIN QUANTUM DOTS

239

Leonard, D., Krishnamurthy, M., Reaves, M. C., DenBaars, P. S., and Petroff. M. P. (1993). Appl. Phys. Lett. 63, 3203. Lipsanen, H., and Sopanen, M. (1995). Phys. Rev. B. 51, 13868. Moison, J. M., Houzay, F., Barthe, F., Leprince, L., Andre, E., and Vatel, 0. (1994). Appl. Phys. Lett. 64, 196. Mukai, K., Ohtsuka, N., Sugawara, M., and Yamazaki, S. (1994). Jpn. J . Appl. Phys. 33, L1710. Mukai, K., Shoji, H., Ohtsuka, N., and Sugawara, M. (1996a). Appl. Phys. Lett. 68, 3013. Mukai, K., Ohtsuka, N., Hajirne, S., and Sugawara, M. (1996b). Phys. Rev. B. 54, R5243. Mukai, K., Sugawara, M. (1998). Jpn. 1. Appl. Phys. 37, 5451. Nakayama, H., and Arakawa, Y. (1994). 7th Int. Con$ Superlattices, Microstructures, and Microdevices, Banff, Canada. Ohnesorge, B., Albrecht, M., Oshinowo, J., Forchel, A., and Arakawa, Y . (1996). Phys. Rev. B. 54, 11532. Oshinowo, J., Nishioka, M., Ishida, S., and Arakawa, Y. (1994). Appl. Phys. Lett. 65, 1421. Raymond, S., Fafard, S., Charbonneau, S., Leon, R., Leonard, D., Petroff, M. P., and Merz, L. J. (1995). Phys. Rev. B. 52, 17238. Raymond, S., Fafard, S., Poole, J. P., Wojs, A., Hawrylak, P., Charbonneau, S., Leonard, D., Leon, R., Petroff, M. P., and Merz, L. J. (1996). Phys. Rev. B. 54, 12548. Sallese, J. M., Carlin, F. J., Gaihanou, M., and Grunberg, P. (1995). Appl. Phys. Lett. 67, 2633. Sercel, P. C. (1995). Phys. Rev. B. 51, 14532. Shoji, H., Mukai, K., Ohtsuka, N., Sugawara, M., Uchida, T., and Ishikawa, H. (1995). I E E E Photo. Tech. Lett. 7, 1385. Sugawara, M. (1998). Prnc. of Optoelectronics '98, Photonic West. San Jose. Tabuchi, M., Noda, S., and Sasaki, A. (1992). Science and Technology of Mesoscopic Siructures (Springer-Verlag. Tokyo, p. 379. Tsang, W. T. (1978). Appl. Phys. Lett. 33, 245. Uskov, A. V., Mcinerney, J., Adler, F., Schweizer, H., and Pilkuhn, M. H. (1998). Appl. Phys. Lett. 72, 58. Utzmeier, T., Postigo, A. P., Tamayo, J., Garcia, R., and Briones, F. (1996). Appl. Phys. Lett. 69, 2674. Vollrner, M., Mayer, J. E., Riinle, W. W., Kurtenbach, A,, and Eberl, K. (1996). Phys. Reu. B. 54, R 17292. Wang, G., Fafard, S., Leonard, D., Bowers, E. J., Merz, L. J., and Petroff, M. P. (1994). Appl. Phys. Lett. 64. 2815. Yu, H., Lycett, S., Roberts, C., and Murray, R . (1996). Appl. Phys. Lett. 69, 4087.

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SEMICONDUCTORS AND SEMIMETALS, VOL. 60

CHAPTER6

Self-Assembled Quantum Dot Lasers Hajime Shoji OPTICAL SEMICONDUCTOR DEV~CES LABORATORY FUJ~TSU LABOKATQRIES LTD. ATSUGI. JAPAN

I. INTRODUCTION. . . . . . . . . . . . . . . . . . . 11. FUNDAMENTAL PROPERTIES OF Q U A N T U M - D O T LASERS . . 1. Gain Characteristics . . . . . . . . . . . . . . . 2. Threshold Current . . . . . . . . . . . . . . . . 3. Dynamic Characteristics . . . . . . . . . . . . . 111. FABRICATION OF SELF-ASSEMBLED QUANTUM-DOT LASERS. 1. Fabrication . . . . . . . . . . . . . . . . . . . 2. Device Characteristics . . . . . . . . . . . . . . 3. Limiting Factors of’Laser Performance . . . . . . . IV. KEYTECHNOLOGIES FOR THE NEXT ERA . . . . . . . . I . Close1.v Slacked Quantum-Dot Lusers . . . . . . . . 2. Columnur Quantum-Dot La~ers . . . . . . . . . . 3. Long- Wavelength Quantum-Doi Lasers . . . . . . . 4. Quantum-Dot Vertical-Cavity Surface-Emitting Lasers . V. CONCLUSION. . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . , . .

.

.

,

,

. .

. . . . . . . . . , , . . , . . , . . . . . . . . . .

24 1 243 244 246 248 249 250 255 261 269 270 213 216 219 282 283 283

I. Introduction The quantum-dot laser is one of the most representative candidates for the practical application of quantum-dot structures. Originally called the quantum-box laser, it was first proposed by Arakawa and Sakaki in 1982. Three-dimensional quantum confinement of electrons and holes in a quantum dot with a size equal to or below the exciton Bohr radius causes the atom-like density of states, or delta function, which is believed to be essential to improved semiconductor laser performance. The effect of three-dimensional quantum confinement was shown in an experiment using a strong magnetic field normal in quantum wells (Arakawa et al., 1983). Semiconduc24 1 Copyright I I Y Y 9 by AcJdemlc Press All ~ i p h l rof reprodurtmn In m y lorm reserved ISBN 0-12-752169-0 ISSN M I R O - X ~ X ~ ,~~3~ 0 0 0

242

HAJIME SHOJI

tor lasers equipped with quantum dots as the active region are expected to exhibit ultralow threshold current, temperature-insensitive operation, narrow spectral linewidth, and large modulation bandwidth (Arakawa and Sakaki, 1982; Asada et al., 1986). All of these features are attributed to the quantum dots’ enhanced gain properties, especially the extremely high differential gain, induced by their three-dimensionally discrete density of states. Despite its theoretically predicted superiority, no quantum dot laser has been actually realized. This is because quantum structures fabricated so far have not produced a strong carrier confinement, especially in a lateral direction. In the last decade, two approaches have been tried for realizing quantum dot structures. The first one is a precise patterning technique (Hirayama et al., 1994), originally developed for quantum-wire structures. It involves patterning of quantum wells by wet or dry etching and overgrowth on the patterned structure, or selective growth on a patterned substrate (Kash et al., 1986; Fukui et al., 1991). Because of recent progress in electron beam lithography and etching techniques, InGaAs-InGaAsP quantum-dot lasers became a reality in 1994 (Hirayama et al., 1994). However, the threshold current density was as high as 7.6 kA/cmZ even at 77 K. In this approach, it is generally very difficult to obain a high-quality structure without damage, since nonradiative recombination, especially at the surface of etched sidewalls where the active region is exposed to air, severely limits laser performance. The second approach is self-organization of quantum dots during highly strained heteroepitaxy, and it has been very successful. Uniform dots with small size, high density, and high emission efficiency, unachievable with conventional techniques, have been easily realized in conventional molecular beam epitaxy (MBE) and metalorganic vapor phase epitaxy (MOVPE). The self-organization technique has lead to three methods for fabricating quantum-dot structures: (1) spontanous formation of In(Ga)As islands on GaAs in the Stranski-Krastanow (SK) mode (Leonard et al., 1993; Kirstaedter et al., 1994; Oshinowo et al.. 1994; Tackeuchi et al., 1995; Mirin et al., 1994); (2) self-organization of dot-like disk structures on (311) B GaAs substrates (Notzel et al., 1994); and (3) self-formation of InGaAs dots on GaAs by alternative supply of InAs and GaAs in atomic layer epitaxy (ALE) (Mukai et al., 1994 and 1996a). Furthermore, the self-organization of quantum dots has been successful in many material systems, such as InAs on InP (Fafard et al., 1996; Nishi et al., 1998); InP on InGaP (Moritz et al., 1996); and GaSb (Hatami et al., 1995) or InSb, GaSb, and AlSb on GaAs (Glaser et al., 1996). Attempts to create a new quantum-dot structure, including development of new material systems and new growth mechanisms, are ongoing, bringing with them a deeper understanding of the growth

243

6 SELF-ASSEMBLED QUANTUM DOTLASERS

mechanism of self-assembled quantum-dot structures. Recent attempts to fabricate quantum-dot injection lasers by the methods mentioned above have been reported by several groups. Kirstaedter et al. (1994) reported the first demonstration of InAs-GaAs quantum-dot laser grown in the SK mode in 1994. Also in 1994, Temmyo et al. reported a continuous-wave (CW) lasing in an InGaAs-AlGaAs quantum-disk structure on (31 l)B GaAs although the lateral size of the disk was 60 nm, which was rather large for sufficient quantum confinement. In 1995, Shoji et al. reported a laser oscillation at 80 K in InGaAs-GaAs quantum-dot lasers grown by ALE. Besides these researchers, several other groups have reported successful laser oscillation (Moritz et al., 1995b; Mirin et al., 1996; Kamath et al., 1996; Shoji et al., 1996a). Clearly, research on the quantum dot structures is creating a new field covering growth techniques and device applications. In this chapter we describe semiconductor lasers with self-assembled quantum dots in the active layer. Section I1 briefly summarizes the fundamental properties of quantum dot lasers from theoretical points of view as well as what can be expected from them. Section I11 introduces actually fabricated quantum-dot lasers, mainly focusing on the results of our work and taking InAs-GaAs quantum-dot lasers -which are most studied as examples. The fabrication process, lasing characteristics and problems to be solved are discussed in detail. Section IV treats several key technologies for improving quantum-dot lasers, including closely stacked quantum-dot structures, columnar quantum-dot structures, and quantum dots emitting in long wavelength region. The chapter ends with a discussion of how these technologies can be applied in quantum dot vertical cavity surface emitting laser (VCSEL). ~

11.

Fundamental Properties of Quantum Dot Lasers

Most of the performance expected from quantum-dot lasers originates in their atom-like density of states. Figure 6.1 shows how active regions in semiconductor lasers and the density of states of electrons vary with an increase in the dimensions of quantum confinement. By the use of multidimensional quantum-confined structures, freedom of carrier motion is reduced to a limited direction. Ideally, the carriers are completely confined in the quantum dot. For that reason, the dot’s density of states is expressed by the delta function. The width-of-gain spectrum is determined only by homogeneous broadening due to the intraband relaxation in the quantum dot. These gain properties are the basis of those features that give the

244

HAJIMESHOJI

Bulk

Quantum well

‘li D(E)

D(E)

Quantum wire

W)

Quantum dot

WE)

FIG. 6.1. Variations in active regions in semiconductor lasers and the density of states for bulk, quantum well, quantum wire, and quantum dot.

quantum-dot laser its advantages over conventional lasers, and we discuss them here. The theoretical treatment of quantum dots is provided in Chapter 1. Basic theories of quantum-dot lasers and conventional lasers are summarized in several papers (for example, Asada et al., 1986; Arakawa and Yariv, 1986). 1. GAINCHARACTERISTICS

The basic theory of the gain characteristics of quantum dots was derived by Asada et al. (1986). This density-matrix theory gives the linear gain of quantum dots without inhomogeneous broadening as follows:

where w is the angular frequency of light; e0 and p0 are, respectively, the dielectric constant and permeability of the vacuum; n, is the refractive index of the quantum dot; f, and J , are the Fermi functions; and M is the dipole moment of the quantum-dot structure. L(E, Q) corresponds to the relaxation

6 SELF-ASSEMBLED QUANTUM DOTLASERS

245

1o4

ox

6

looxlooxloo A3

v

C .-

103

2x102

1.2

1.3

1.4

1.5

1.6

1.7

Wavelength, h (pm) FIG. 6.2. Calculated gain spectra at 300K for a quantum dot (box), quantum wire, quantum well (film), and bulk. (Reprinted with permission from Asada et al., 1986. $'I 1986 IEEE.)

broadening given by the Lorentzian function, where T~,,is the intraband relaxation time. D(E) is the density of states of electron-hole pairs with the same energy levels, where d,, d,, and d, are the dot sizes assuming a box shape, respectively, and 6(E) is the delta function. The density of states in quantum dots is illustrated in Fig. 6.1. Densities of electrons and holes in quantum dots are determined by the neutral condition. Figure 6.2 shows calculated gain spectra at 300K for quantum dot (box), quantum wire, quantum well, and bulk structures at a constant ~ . for the bulk structure, the carrier density, N , of 3 x 10l8~ m - Except thickness in the quantum-confined direction was set at 100A.We clearly see that enhancement of peak material gain and narrower gain bandwidth in the quantum-dot structure are the results of an increase in the dimension of quantum confinement from bulk to quantum dot. The peak gain of the quantum dot increases by one order of magnitude compared with that of the quantum well. This result also indicates that the differential gain, dg/dN, is drastically improved in the quantum dot due to the large peak gain and the narrow gain bandwidth.

246

HAJIMESHOJI

2. THRESHOLD CURRENT The large optical gain and differential gain result in a low threshold current operation. Assuming that the current is injected only through the top surfaces of cubic quantum dots, as described by a schematic image shown in Fig. 6.3, the relation between the injection current and the carrier density is expressed by a simple formula as follows:

J

=

td,eN/z,

(6.4)

where J is the averaged current density defined by dividing the total current by the whole area of the active region ( J = I/WL in Fig. 6.3); e is the electron charge; N is the carrier density; z, is the carrier lifetime; and 5 is the filling factor of quantum dots in the active region. Here, the active region is assumed to consist of a single dot layer. Referring to Fig. 6.3, 5 is defined as 5 = d,d,/WL, where W is width of active region and L is the cavity length. The effect of interaction among dots and finite potential depth is ignored. Generally, the carrier lifetime z, consists of radiative and nonradiative components of z, and znr, respectively, as (6.5)

Current, I

FIG. 6.3. Schematic of a quantum-dot laser.

6 SELF-ASSEMBLED QUANTUM DOTLASERS

241

The radiative lifetime is given approximately by the bimolecular lifetime as

zr

=N

/[

c [: I M I ~ E ~ Lmw) U 4

3/2Eolj2n , ~ n h - 4 imn

-

(6.6)

Using Eqs. (6.1) through (6.6), the optical gain can be related to the current density. Figure 6.4 shows calculated peak gains as a function of current density for four types of lasers with active regions of quantum dots, quantum wires, quantum wells and bulk (Asada et al., 1986). The structures assumed here are the same as in Fig. 6.2, which shows the increase of peak gain with the increased dimension of quantum confinement. The threshold gain, g t h , is denoted as follows by considering the optical confinement factors in the given structures:

Current density, J (A/cm2) FIG. 6.4. Calculated peak gain at 300 K for a quantum dot (box), quantum wire, quantum well (film), and bulk. The dashed line on each curve indicates the threshold gain. The corresponding threshold current density is also shown. (Reprinted with permission from Asada et al., 1986. 1Ci 1986 IEEE.)

248

HAJIME SHOJI

where u,, and a,, are the absorption losses in the active region and other regions, respectively; tx,F,, and 5, are the optical confinement factors in the x,y , and z directions, respectively; and R is the facet reflectivity. 5, and 5, can be approximately defined as the filling factors in the x and z directions, respectively, because the optical field in the cavity is much larger than that in the quantum dot. t, is determined by the waveguide structure. For example, assuming that tx= 5, = 0.5 for a quantum-dot structure, 4, = 1 and t, = 0.5 for a quantum-wire structure, 5, = tz= 1 for quantum-well and bulk structures, and assuming reasonable values for other parameters (Asada et al., 1986), the threshold gains can be calculated by Eq. (6.7). Then, if the results are put into Fig. 6.4, the threshold current densities for the given structures are derived. As noted in Fig. 6.4, lower threshold current density should be achieved in quantum-dot lasers-about one tenth that of quantum well lasers. Of course. this value strongly depends on the parameters assumed, and they should be much improved by optimizing the structure. The low threshold current in the quantum-dot laser also brings about a lesser temperature dependence of the threshold current due to strong confinement of the electrons and holes. Regarding threshold gain, there is another attractive feature of the quantum-dot laser. As described above, the internal loss expressed by the first and second terms of Eq. (6.7) becomes smaller in the quantum-dot laser, therefore, larger differential quantum efficiency, given by

r 1 InRf

1

is expected, where qd is the differential quantum efficiency, qi is the internal quantum efficiency, and R, and R , are the reflectivities of the front and rear facets, respectively.

3. DYNAMIC CHARACTERISTICS Large differential gain results in improvement of dynamic characteristics. The relaxation oscillation frequency, L, which determines the intrinsic modulation bandwidth, is given by f

' =L/ii$g 2a

(6.9)

6 SELF-ASSEMBLED QUANTUM DOTLASERS

249

where neff is the effective refractive index of the waveguide, c is the speed of light in the vacuum, Po is the photon density in the cavity, and zp is the photon lifetime. It is quite clear that greater differential gain results in a larger relaxation oscillation frequency, which potentially enhances the modulation bandwidth of the quantum-dot laser. Narrower spectral linewidth can be also expected in the quantum-dot laser. The spectral linewidth, Av, is expressed by

(6.10) (6.11) where M , R,, vg, hw, r, nsp, n, and Po are, respectively, the linewidth enhancement factor, the mirror loss, the group velocity of light, the photon energy, the optical confinement factor, the spontaneous emission factor, the effective refractive index, and the output power. A large differential gain results in a linewidth enhancement factor of nearly zero, leading to smaller spectral linewidth. These parameters are strongly related to the direct modulation characteristics, and chirping under high-speed modulation should be suppressed because of the small a, leading to long transmission distance in fiber communication applications.

111. Fabrication of Self-Assembled Quantum-Dot Lasers

In the realization of self-assembled quantum-dot lasers, quantum-dot structures grown in the SK mode are the most successful to date for device performance. Recent intensive work has clarified the properties of uctuul quantum-dot lasers grown in the SK mode. Most of this work focuses on InAs or InGaAs quantum dots on GaAs because they are suitable for laser application, given their small diameter of 20 nm, high density of > 1 x 10'' cm ',typical size distribution of lo%, high emission efficiency, and so on. Room-temperature CW lasing has been demonstrated in In(Ga)AsGaAs quantum dots. This section describes the fabrication and lasing characteristics of selfassembled In(Ga)As-GaAs quantum-dot lasers, focusing mainly on the results of our work. Also discussed are several factors that limit the performance of quantum-dot lasers and what can be done about them. ~

- -

250

HAJIMESHOJI

1. FABRICATION Formation of Quantum Dots in Stranski-Krastanow Mode

The SK growth mode is used mostly in the formation of quantum-dot structures. Typically, formation of In(Ga)As-GaAs quantum dots in this mode occurs at temperatures between 460°C and 550°C. The formation is reported both in molecular beam epitaxy (MBE) (Leonard et al., 1993; Oshinowo et al., 1994; Tackeuchi et al., 1995; Mirin et al., 1995; Sugiyama et al., 1996) and in metalorganic vapor phase epitaxy (MOVPE) (Ledentsov et al., 1996; Heinrichsdorff et al., 1996 and 1997). A few-monolayer supply of In(Ga)As on GaAs result in the formation of three-dimensonal islands on an extremely thin film called a wetting layer. The typical lateral diameter obtained by MBE and MOVE ranges between 12 nm and 20 nm, and the dot density is typically between 2 x 10'' cm-2 and 1 x 10" cm-2. Details on the SK growth mode are given in Chapter 3. Optimization of dot structure is important in semiconductor lasers. Too small a size results in the absence of a quantum-confined level or in a small energy difference between the barrier and the quantized energy level of dots. Too large a size leads to a weaker quantum size effect, which means that the atom-like nature of the dots will not be extracted. Dot density is another important factor. For example, although it depends on the waveguide structure, if there is no fluctuation in dot size, the optimum areal coverage in a single active plane will be about a few percent to minimize the threshold current (Asada et al., 1986). However, in actual quantum dots, there is a large size fluctuation. Higher-density dots are required to obtain large modal gain; low-density dots lead to gain saturation due to small optical confinement to the active layer, which degrades performance (Schmidt et al., 1996). Therefore, in addition to increasing in-plane density, vertically stacked dot structures are often required. Figure 6.5(a) and (b) shows, respectively, a plan view and a cross-sectional view, taken with a transmission electron microscope (TEM), of a triplestacked quantum-dot structure (Shoji et al., 1997a). The grown quantum dots were 20 nm in diameter and 5 nm in height. The areal coverage was about 10%. Each dot layer was separated by GaAs barriers that were 20 nm thick. In this case, the neighboring dot layers in the vertical direction could be optically coupled through the waveguide structure, but they were not electronically coupled because the barrier layer between the dot layers was thick enough. Stacked dot structures with 10 or more layers have also been reported (Saito et al., 1996; Pan et al., 1997). Electronically coupled quantum-dot structures with thinner barriers are discussed in Sections IV. 1 and IV.2.

6 SELF-ASSEMBLED QUANTUM DOTLASERS

251

FIG.6.5. (a) Plan-view and (b) cross-sectional TEM images of a triple-stacked quantumdot structure. The grown quantum dots were 20 nm in diameter and 5 nm in height. The areal coverage was about 10%. Each dot layer was separated by 20-nm-thick GaAs barriers. (From 1997 IEEE.) Shoji et al., 1997a.

c,

Optical Proper ties

An example of a photoluminescence (PL) spectrum at 300K for a triple-stacked quantum dot structure is shown in Fig. 6.6 (Shoji et al., 1997a). An emission peak at 1.13 pm was clearly observed at room temperature, and the full width at half maximum (FWHM) of the peak was about 80 meV. In most cases, the emission peak of In(Ga)As-GaAs quantum dots is between 0.95pm and 1.2pm. The FWHM of the emission peak represents the uniformity of dot size, and is typically 40-100meV. The

-

252

HAJIMESHOJI Energy (eV) lo-' ground Wetting

1 o'6 800

900

2nd

1000

1100

1200

1300

Wavelength (nm) FIG. 6.6. Photoluminescence spectrum at 300 K for a triple-stacked quantum-dot structure. The emission peak at around 1.13pm is from the ground state, and the full width at half maximum of the peak is about 80meV. (From Shoji et al., 1997a. 0 1997 IEEE.)

narrowest linewidth is 15meV (Oshinowo et al., 1994). This value is determined by inhomogeneous broadening, not by homogeneous broadening. Emission linewidth of a single quantum dot is much less than 1 meV at low temperature (Grundmann et al., 1995), which corresponds to the phase coherence time in a quantum dot. Consequently, if the inhomogeneous broadening dominates the spectral linewidth, almost constant FWHM is obtained over a wide temperature range. This fact is in great contrast to cases of bulk material and quantum wells. In Fig. 6.6, the asymmetric PL spectrum suggests the existence of several excited states. By the Gaussian fitting to the PL spectrum, the second state of electrons is found at around 1.04- 1.05pm. The energy difference between the ground state and the second state is about 90meV, which agrees well with a theory that considers the grown dot size. Although the third state is not identified clearly in the PL spectrum, assuming that the energy level between the neighboring states is equally spaced in the parabolic potential (Mukai et al., 1994 and 1996b; Ahopelto et al., 1995), the third level should be at around 0.97 pm. Note that a peak at 0.92 pm is from the underlying wetting layer, and a peak at 0.88 pm is from the GaAs barrier layer. The number of excited states is strongly affected by dot size. Figure 6.7 illustrates this relationship. If the dot is small enough ( < 15 nm), only the transition between the ground states of electrons and holes become a dominant recombination process. Although emission peaks from the excited

-

253

6 SELF-ASSEMBLED QUANTUM DOTLASERS Dot size

Small dot

Large dot

Energy

El E2 E3

El

Allowed transition

Ground state only

Electroluminescence

kE Ei

Several excited states

Ei Ez E3

E

FIG. 6.7. Effect of dot size on the excited states of electrons and holes.

states are also generally observed together with those from the ground state because of nonorthogonality due to anisotropy of dot shape, the emission from the ground state always becomes dominant even with increasing excitation. On the other hand, in larger dots, there are several excited states of electrons in the conduction band, which leads to the transitions from the excited states of electrons and holes. The result in Fig. 6.6 corresponds to the latter case.

Device Fabrication A schematic view of a typical In(Ga)As-GaAs quantum dot laser is given in Fig. 6.8. Profiles of the conduction-band energy and the corresponding refractive index are depicted. The active layer of a quantum-dot laser consists of a two-dimensional or three-dimensional array of quantum dots surrounded by a pair of higher bandgap barriers to effectively confine injected carriers. The active layer is sandwiched by lower and upper cladding layers with a higher band gap energy (smaller refractive index), resulting in the formation of an optical waveguide. Thus, the structure is basically the same as that of conventional double heterostructure lasers. Ridge-waveguide structure is often employed to evaluate the performance of quantum-dot lasers in addition to broad-area lasers. A schematic structure of a fabricated ridge-waveguide quantum-dot laser is shown in Fig. 6.9

254

HAJIMESHOJI

mAlGaAs c-GaAs substrate cladding layer

GaAs barrier

pAlGaAs pGaAs cladding layer contact layer

Refractive

t

index

FIG. 6.8. Schematic of a typical In(Ga)As-GaAs quantum-dot laser, showing profiles of the conduction-band energy and the corresponding refractive index. The dashed curve shows the optical field in the lateral direction.

FIG. 6.9. Schematic of a fabricated ridge-wave guide quantum-dot laser. The laser contains a triple-stacked quantum-dot structure. The width of the ridge-waveguide is about 2.5 pm.

6 SELF-ASSEMBLED QUANTUMDOTLASERS

255

(Shoji et al., 1997a). The laser structure was grown on a (001) n-GaAs substrate followed by a 0.9-pm-thick n-Al,,,Ga,.,As cladding layer, a quantum-dot active layer, a 1.2-pm-thick p-Al,,,Ga,,,As classing layer, and a 0.3-pm-thick p-GaAs contact layer. As the active layer, a triple-stacked quantum-dot structure was employed to obtain larger optical confinement to the active layer. In this case, the optical confinement factor to the active-layer plane was about 6% (actual optical confinement to quantum dots is much less than this value because of low areal coverage of quantum dots in the active layer). A narrow ridge-waveguide structure is usually formed by chemical or dry etching of the p-GaAs contact layer and a part of the p-AlGaAs cladding layer. In Fig. 6.9, the width of the ridgewaveguide was about 2.5 pm, which enabled stable control of the lateral mode. After growing the entire structure, several laser chips with different device parameters were prepared for measurement. Typically, cavity length and facet reflectivity varied case by case. In addition, the number of dot layers often varied wafer to wafer.

2. DEVICE CHARACTERISTICS

Electroluminescence Electroluminescence (EL) from quantum dots under current injection below the threshold gives us much information on the subbands of quantum dots as well as the PL spectrum. Figure 6.10 shows measured EL spectra for several injection currents (Shoji et al., 1997a). Clear discrete peaks were found in the EL spectra which were attributed to subbands of quantum dots (Ahopelto et al., 1995; Mukai et al., 1996a; Grundmann et al., 1996; Xie et al., 1996). The three peaks in Fig. 6.10 corresponded to the transitions between the ground states, the second states, and the third states of electrons and holes, respectively, in descending order of wavelength. Note that a shoulder peak at around 0.9 pm originated from the wetting layer just below the quantum dots. In Fig. 6.10, with an increasing injection current, the emission peak shifts from the longer wavelength side to the shorter wavelength side. This is mainly due to the band-filling effect. The stronger emission at higher-order subbands directly reflects the larger degeneracy of eigenstates at them. There are more states to be filled with carriers, at these subbands, which leads to a stronger emission. Slight shifts of the peaks to the shorter wavelength side with an increasing injection current are mainly due to the overlap of the emission from the higher-order subbands. Because the fluctuation in dot size and composition causes the large broadening of each peak, the peak

256

HAJIMESHOJI

L = 300 ;m as-cleaved

3rd 2nd

50?*

800

I

1

1000 1100 Wavelength (nm)

900

1200

FIG.6.10. Electroluminescence spectra for several injection currents, showing clear discrete peaks corresponding to the subband of quantum dots. The peak in the longest wavelength side is from the ground state. (From Shoji et al., 1997a. C 1997 IEEE.)

wavelength of the lower-order subbands is shifted to the shorter wavelength side by the increased emission from the higher-order subbands. If a quantum-dot structure with a smaller size fluctuation can be fabricated, this kind of peak shift will vanish.

Lasing Churucteristics Laser oscillation at room temperature has been achieved in self-assembled In(Ga)As-GaAs quantum-dot lasers by several groups (Kirstaedter et al., 1994; Mirin et al., 1996; Kamath et al., 1996; Shoji et al., 1996a and b). An example of cavity-length dependencies of threshold current density for lasers with different numbers of dot layers and different facet reflectivities is shown in Fig. 6.11 (Shoji et al., 1997a). Here, the threshold current density was normalized by a value for one dot layer for comparison. The open circles represent as-cleaved devices, and the closed circles represent devices with high-reflection (HR) coating on both facets. As shown in the figure, the reduced cavity loss by the use of long cavity structures and the enlarged optical gain by multiple stacking of the dot layers led to the reduction of the threshold current density. A 95%-HR coating also greatly reduced the threshold current density, especially in short cavity lasers. In 600-pm-long and 900-pm-long lasers with triple-stacked dot layers and HR-coated facets,

6 SELF-ASSEMBLED QUANTUM DOTLASERS

300 K, pulsed

I

257

I

I

300

600

900

1200

Cavity length (pm) FIG. 6.11. Cavity-length dependence of threshold current density for quantum-dot lasers with different numbers of dot layers and different facet reflectivities, Niayeris the number of quantum-dot layers in the active layer. The threshold current densities are normalized for one dot layer for comparison. (From Shoji et al., 1997a.)

the threshold current densities for a dot layer were smaller than 1 kA/cm2. In these lasers, room-temperature CW operation was achieved (Shoji et al., 1996b). Figure 6.12 shows light output versus current characteristics for a CW-lasing quantum-dot laser. In this case, the threshold current was about 140 mA at 20"C, which corresponded to the threshold current density of 3.1 kA/cm2. Much lower threshold current density, below 400 A/cm2 at room temperature, has also been reported (Kamath et al., 1997b; Heinrichsdorff et al., 1997). The difference in the threshold current density is mainly due to the difference in the emission efficiency of quantum dots. Reduction of nonradiative recombination as a result of improved crystal quality leads to the improved emission efficiency, especially at room temperature, which results in lower threshold current density. Various parameters are derived from the mirror loss dependence shown in Fig. 6.11. The internal loss, m i , the differential quantum efficiency, yld, and the internal quantum efficiency, q i are important factors in laser quality. Although these parameters strongly depend on the waveguide structure and the quality of the active region, those values can be derived from a relationship expressed by Eq. (6.8). Experimentally, the internal loss, ( x i , was estimated to be 3-5 cm-' (Shoji et al., 1997a). Of course, this value might depend on the density of quantum dots in the active layer. The internal

258

HAJIME SHOJI 1.o

z v

c

N,aye, = 3,L = 900 pm HR-coated 0.8 20°C

cw

0.6

3 Q c

3

E0)

0.4

7

0.2

a

100

200

300

400

Current (mA) FIG. 6.12. Light output versus current characteristics for a quantum-dot laser under the CW condition. Triple-stacked quantum dots are in the active layer (PIloyer = 3). and the cavity length is 900pm ( L = 900pm). The threshold current is about 140mA at 20’C, which corresponds to the threshold current density of 3.1 kA/cm2.

quantum efficiency is still lower than that of well-established quantum-well lasers, which have an internal quantum efficiency of nearly 100%. The value of quantum-dot lasers is still between 40% (Bimberg et al., 1997) and 70% (Ustinov et al.., 1996). A large amount of nonradiative recombination at room temperature might be the cause of the small value. The phonon bottleneck, discussed in Chapter 5, might also degrade the internal quantum efficiency. The gain properties of quantum dots are influenced by spectral characteristics dependent on device parameters such as number of dot layers and cavity loss. Figure 6.13 shows a systematic comparison of wavelength behaviors of quantum-dot lasers with different device parameters (Shoji et al., 1996a, 1996b, and 1997a). It indicates clearly that increased optical gain in the multi-stacked dot layer and decreased cavity loss suppressed the emission from higher-order subbands, resulting in lasing at lower-order subbands. In a comparison using lasers with a single dot layer, when the cavity length was 300 pm long, lasing occurred at the wetting layer, while in a 900-pm-long laser with smaller cavity loss, lasing occurred at the thirdorder subband. In the same manner, in a comparison using 900-pm-long lasers, the increase in dot layers from one to three led to second-state lasing from third-state lasing. Further reduction of the cavity loss by the HR coating on both facets of the laser finally enabled CW lasing at the ground

6 SELF-ASSEMBLED QUANTUM DOTLASERS

rayer =3 = 900 pm

259

175 mA (CW) ground state

WHR)

1 50

20 10

5

550 rnA (pulsed 2nd state

layer = 3

100

20

1 = 900 pn

layer'

360 mA (pulsed) 3rd state

/aver = 1

-300 vm

800

900 1000 1100 Wavelength (nm)

1200

FIG.6.13. Wavelength behaviors of quantum-dot lasers with different device parameters. The increased optical gain in the multi-stacked dot layer and the decreased cavity loss suppressed the emission from higher-order subbands, resulting in lasing at the lower-order subbands. In a laser with triple-stacked dot layers (Nlayer = 3) and a 900-pm-long cavity. CW lasing at the ground state was observed. (From Shoji et a]., 1997a. $? 1997 IEEE.)

state. A key point here is that the lasing wavelength did not change continuously from the higher-order subbands to the lower-order subbands, but changed discontinuously, reflecting the existence of discrete quantumconfined subbands. A relationship between the lasing wavelength and the threshold gain for one dot layer, as shown in Fig. 6.14, is evidence that the lasing wavelength jumps from the second state to the ground state with a lowering of the threshold gain for one dot layer. The critical threshold gain

260

HAJIME SHOJI

.

0

' 2

-

I

4

1

.

6

8

1

0

Threshold gain for one dot layer (cm-I) FIG 6 14. Relationship between the lasing wavelength and the threshold gain for one dot layer. The lasing wavelength jumps from the second state to the ground state when the threshold gain for one dot layer is lowered by the use of a long cavity structure and a multiply stacked dot layer (From Shoji et al. 1997a ( 1997 IEEE)

for one dot layer was found to be 3.5 cm - in this case, as marked by a dashed line (Shoji et al., 1997a). A relationship between modal gain and current density can be also derived from the cavity-loss dependence of threshold current density, as shown in Fig. 6.15 (Shoji et al., 1997a), where both values were normalized for one dot layer. It was found that the differential gain for the current density depended on the subband at which lasing occurred. The different differential gains can be attributed to the variation in degeneracy between the ground state and the second state-the larger degeneracy at the higher state leads to the larger differential gain. This means that lasing at the ground state reduces the threshold gain for one dot layer by multiply stacking the dot layer, increases the optical gain for one dot layer by increasing the in-plane dot density, or reduces the cavity loss by the use of a long cavity structure and HR coating. At around the cross-point of the two gain curves, which coincides with the critical threshold gain of 3.5 cmin Fig. 6.14, lasing occurred at the two subbands at the same time because of comparable modal gains at the two subbands at a given current density. Measurement of material gain was also carried out (Kirstaedter et al., 1996). A large gain of 7 x lo4cm- was reported at a current injection level of only 80 A/cm2. Another important feature observed in actual quantum-dot lasers is gain saturation (Schmidt et al., 1996). For example, as shown in Fig. 6.13,

-

6 SELF-ASSEMBLED QUANTUM DOTLASERS

0.1

261

10

1

Current density for one dot layer (kA/crn2)

FIG. 6.15. Relationship between the modal gain and the current density derived from the cavity-loss dependence of the threshold current density, where both values were normalized for one dot layer. It was found that the differential gain for the current density depended on the 1997 IEEE.) subband at which lasing occurred. (From Shoji et al., 1997a.

c?

although CW lasing at the ground state was achieved in a 900-pm-long HR-coated laser, the lasing wavelength of 1105 nm was about 25 nm shorter than the ground-state PL peak wavelength of 1130nm. This is the gain saturation effect in quantum dots. In quantum-dot structures with large size and composition fluctuations, only the dots with relatively high density can contribute to lasing because gain from a single quantum dot is almost constant. Consequently, carrier density in low-density dots is not clamped even at the threshold of the high-density dots, and the carriers begin to distribute to the second state after the ground state has been filled with carriers, resulting in the generation of optical gain at the tail of the second state. This situation is essentially different from that of conventional bulk and quantum-well lasers. The total optical gain takes the maximum value at a wavelength shorter than the PL peak. In fact, a lower threshold gain, as shown in Fig. 6.14, led to a longer lasing wavelength because carrier distribution to the higher-order subbands was suppressed. The gain saturation effect sometimes appears in light-current characteristics. Even once lasing starts at the ground state just above the threshold current, if the optical gain at the second state is close to the gain at the ground state, current injection induces the increase in optical gain at the second state due to the gain saturation at the ground state, resulting in the beginning of lasing at the second subband. This behavior is clearly shown in Fig. 6.16. At the threshold current, only a sharp peak was observed in the lasing spectrum. However, the spectrum was broadened because many dots

262

HAJIMESHOJI

2nd

j

Ground

state \ state

1

20 C Pulsed

A,:

'

A

ncn

Current (A)

1000 1050 ..-.?length (nm)

FIG. 6.16. Large kink observed in their light-current characteristics of a quantum-dot laser. Because of gain saturation at the ground state, the lasing spectrum broadened with an increase in the injection current, and lasing at the second state began. At that point a large kink in the light-current curve was observed.

began to lase with increased current injection; lasing finally began at the second state due to the gain saturation at the ground state. For that reason, a large kink was observed in the light-current curve at which lasing at the second state started.

Temperature Depetzdence

Temperature dependence of lasing characteristics is of great interest both practically and physically. Temperature-dependent threshold characteristics and spectral characteristics are popular targets of research. In Fig. 6.17, the temperature dependency of lasing wavelength for two lasers are compared, together with that of PL peak wavelength (Shoji et al., 1997a and b). The two lasers lased at different subbands at room temperature. One oscillated at the ground state (type-A; 900-pm-long HR-coated laser); the other oscillated at the second state (type-B; 900-pm-long ascleaved laser). With a lower temperature, interesting changes were observed in the laser's spectral characteristics. Type A's lasing wavelength gradually

6 SELF-ASSEMBLED QUANTUM DOTLASERS

263

Type B, as-cleaved

1000 . '

50

'

.' .. .'. .. '. . ..' ... 100 150 200 250 300 '

'

'

Temperature (K) FIG. 6.17. Temperature dependencies of lasing wavelength for two lasers together with that of the PL peak wavelength. The two lasers lased at two different subbands at room temperature; one oscillated at the ground state (type A, 900-pm-long HR-coated laser), and the other oscillated at the second state (type B, 900-pm-long as-cleaved laser). With a lowering temperature, interesting spectral behaviors were observed. (From Shoji et al., 1997a. (" 1997 IEEE.)

shifted to the shorter wavelength side in proportion to the temperature dependence of the band-gap energy. At the same time, the wavelength difference between the PL peak and the lasing wavelength became smaller, converging at the PL peak when the temperature went below 100K. In contrast, type-B's lasing wavelength jumped to the longer wavelength side at around 260 K. Lowering the temperature increased the emission efficiency and reduced the thermal excitation of carriers to the second state. This allowed lasing at the ground state at the longer side of the second state. Further lowering of the temperature caused the lasing wavelength to converge at the PL peak as in type A. This result reveals again that the two lasers were oscilating at different states at room temperature. A similar result was reported by others (Bimberg et al., 1996, Heinrichsdorff et al., 1997). This comparison of temperature dependence indicates that smaller threshold gain and less distribution of carriers to higher-order states lead to a smaller difference between the lasing and PL wavelengths. This connection fits well with the relationship between lasing wavelength and threshold gain shown in Fig. 6.14.

264

HAJIMESHOJI

Figure 6.18 shows the EL and lasing spectra of the type-A laser in Fig. 6.17 for three temperatures: 60, 200, and 300 K (Shoji et a]., 1997a and b). Distinct differences were observed in the spectra. At 300 K, the threshold current was large and the EL spectra below the threshold were asymmetrically shaped. This indicates that the injected carriers distribute to the second state because of high threshold carrier density, which is caused by poor emission efficiency of quantum dots and large thermal excitation of carriers from the ground to the second state at room temperature. Consequently, the EL peak, which coincided with the PL peak at a low current-injection level, exhibited a large shift to the shorter wavelength side with an increase in current. On the other hand, with a lowering of temperature, the FWHMs of

J/aver = 3,L = 900 pm iR-coated 300 K 150. 100.

900

1000 1100 1200 Wavelength (nm)

FIG.6.18. EL and lasing spectra of the type-A laser from Fig. 6.17 for three temperatures of 60, 200, and 300 K. A change in the carrier distribution to higher-order states was seen in the temperature-dependent spectra. Multi-wavelength of operation peculiar to the quantumdot laser was observed at 60 K. (From Shoji et al., 1997a. (c) 1997 IEEE.)

265

6 SELF-ASSEMBLED QUANTUM DOTLASERS

the EL spectra decreased as the threshold current decreased, and the emission from the second state was well suppressed. The EL spectra at 60 K had almost symmetrical shapes, and the peak wavelength of the lasing spectrum almost coincided with the PL peak. Because of the large decrease in the threshold carrier density and the thermal excitation of the carriers, the effect of higher-order states became negligible, and only the gain at the ground state contributed significantly to the lasing. Interestingly, the lasing at many wavelengths was sometimes observed at 60K, as shown in Fig. 6.18. This was not the multi-mode oscillation observed in ordinary Fabry-Perot lasers. Many quantum dots with different sizes and compositions were lasing at the ground state at the same time because of the reduced threshold carrier density and the increased emission efficiency at 60 K. In Fig. 6.18, the wavelength span of lasing at 60 K was as wide as 50 nm, which was much wider than that of the Fabry-Perot lasers. However, the span decreased as in the lasing spectrum at 200K which is shown in Fig. 6.18. This result clearly shows that the quantum dots are spatially separated in the active layer but optically coupled through the waveguide structure. It also suggests that improved dot uniformity will result in lasing in the narrow wavelength span, leading to an extremely low threshold current, as theoretically predicted. Figure 6.19 shows an example of the temperature dependency of threshold

1 t

zE

.mm rn a -

.PL

R

v

\

Type-8 (as-cleaved)

0 41K

A

7'

h

.-cn

I -

C

3

x .-c cn C

a,

c

50

100 150 200 250 Temperature (K)

300

FIG.6.19. Temperature dependence of the threshold current for type-A and type-B lasers from Fig. 6.17 together with that of the PL intensity. (From Shoji et al., 1997a. CI 1997 IEEE.)

266

HAJIMESHOJI

currents for type-A and type-B lasers, as already shown in Fig. 6.17 (Shoji et al., 1997a and b). Temperature dependence of P L intensity is also shown for comparison. The type-A laser exhibited a high characteristic temperature, To, defined by

);(

I , , x exp

(6.12)

of over 300K in the temperature range below 200K, and the threshold current density was less than 600 A/cm2. In particular, the characteristic temperature was as high as 477 K in the temperature range of 60- 110 K, and the threshold current density was less than 450 A/cm2. However, in the temperature region over 200 K, the characteristic temperature rapidly decreased to a value below 100 K . It was as small as 48 K at room temperature. As for the type-B laser, although it also exhibited a high characteristic temperature of over 400K in a temperature region below 100K, the characteristic temperature started to decrease at 170 K, which was a lower temperature compared with that in the type-A laser. The reason for this unusual behavior of the threshold current at temperatures of 120-130 K ha5 not been determined. Another unusual behavior in the temperature range of 220-260 K -the wavelength shift between the ground state and the second state observed at around 260K, shown in Fig. 6.17-might be attributed to this phenomenon. As for the value of the characteristic temperature, a much higher value of 530 K over a temperature range of 80-220 K has been reported by Bimberg et al. (1997). It can be assumed that the temperature dependence of the threshold current is strongly related to that of the P L intensity; that is, the decrease in the PL intensity with increasing temperature is due to the increased nonradiative recombination in the quantum dots, which results in the proportional increase in the threshold current. However, the results shown in Fig. 6.19 do not necessarily support this assumption. The PL intensity begins to decrease at 150 K, whie the rapid increase in the threshold current starts at 170 K in the as-cleaved laser and at 200 K in the HR-coated laser. Although it is qualitatively correct that a decrease in the P L intensity leads to an increase in the threshold current, the relationship between the two values is not so simple. Further investigation should be done on temperature characteristics, because the energy separation between the subbands is also related to them; nevertheless, the energy separation between the ground state and the second state is much larger than kT of 26 meV, even at room temperature. On the other hand, the comparison between the HR-coated laser and the as-cleaved laser shows that a lower threshold current leads to better temperature characteristics in a higher temperature region. Reduction

6

SELF-ASSEMBLED QUANTUM DOTLASERS

267

of the threshold current is essential in improving temperature characteristics. Also critical is optimizing the energy separation between the subbands of quantum dots as well as improving emission efficiency. Modulation Characteristics As described in Section 11, enhanced differential gain due to the atom-like density of states in quantum dots is expected to bring about excellent modulation in quantum-dot lasers. There have been several studies on the frequency response of quantum-dot lasers through the measurement of direct modulation bandwidth by current injection and the measurement of relative intensity noise (RIN). In the small signal modulation, a 3-dB bandwidth, f3dBr of 7.5 GHz and 8.5GHz has been measured using lasers lasing at the ground state of quantum dots. This modulation bandwidth corresponds to the differential gain of 2-6 x 10- l4 cm2, which is higher than the best values reported for multiple quantum-well lasers (Kamath et al., 1997a and b). Measurement of RIN also gives much information on the dynamic properties of quantum-dot lasers because this measurement is free from the C R constant dependent on device structures. In this case, lasers to be measured are operated under the CW condition, and an observed peak frequency in the noise spectrum corresponds to the relaxation oscillation frequency, f ; . The 3-dB bandwidth of the modulation response is related to the relaxation oscillation frequency by fSdB = 1.55f,. With this method, a of relaxation oscillation frequency of 5.3 GHz corresponding to ,f3,, 8.2 GHz has been reported (Ma0 et al., 1997). As described above, modulation bandwidth or resonant oscillation frequency of over several tens of GHz has been theoretically predicted for an ideal quantum-dot laser. The possible reason for this is twofold: (1) insufficient uniformity of dots might fail to fully extract the potential wide bandwidth, and (2) a relatively long relaxation time from the upper states to the lower states in quantum dots -the so-called phonon bottleneck might limit the bandwidth. Although both factors should contribute to the bandwidth, it is not clear at present which is the dominant factor. Further investigation of carrier dynamics in quantum dots is needed. ~

FACTORS OF LASERPERFORMANCE 3. LIMITING Room-temperature CW operation at the ground state of quantum dots can now be realized through use of self-assembled structure. The lasing

268

HAJIMESHOJI

characteristics are reaching those of well-established quantum-well lasers, especially in threshold current density, However, the obtained performances are still far from theoretical predictions (Arakawa and Sakaki, 1982; Asada et al., 1986). In order to realize the predicted performance in quantum-dot lasers, drastic improvement in fabrication technologies and deep insight into the physics of quantum dots are required. The key improvements necessary are as follows: 0 0

0 0

Dot uniformity, density, and size Crystal quality Phonon bottleneck Emission wavelength

Precise control of dot size and composition is critical for proportional enhancement of optical gain in the active layer. Nonuniformity measured through FWHMs of the PL spectra should greatly reduce. A PL spectrum with an FWHM below lOmeV, which is comparable to or narrower than that of a typical quantum well, is preferable for extracting the atom-like nature of quantum dots. Optimization of dot density and size is also required. Optimum density should be based on dot uniformity. If the uniformity is not sufficient, we need to increase the density, whereas a low density is sufficient for high uniformity. Dot size is critical for temperature-insensitive operation in a high temperature range. If the dot is large, it contains several excited states. A small energy difference between the excited states leads to carrier distribution to the upper states, resulting in an increase in threshold current and poor temperature characteristics. In contrast, if the dot is small, only the ground state can exist in it. In this case, however, the energy difference between the energy levels of the barriers and the quantized states becomes a key parameter. A wider bandgap barrier might also be required. As discussed in the following section, vertically coupled quantum-dot structures work well in controlling dot uniformity and size. For improved device performance, crystal quality also needs to be improved. As shown in Section 111.2, the internal quantum efficiency of quantum-dot lasers is typically less than 80%, which is inferior to that of quantum-well lasers. At present, one of the major causes of poor internal quantum efficiency is insufficient crystal quality. As shown in Fig. 6.19, threshold current tends to increase when PL intensity decreases. It seems certain that the existence of nonradiative centers inside or outside of quantum dots degrades emission efficiency, especially in the room temperature region. Although further investigation is required to identify the origin of the nonradiative center, an improved growth method would lead to

6 SELF-ASSEMBLED QUANTUM DOTLASERS

269

enhanced performance. In fact, excellent threshold characteristics are achieved in lasers with defect-free InAs-GaAs quantum dots grown by MOVPE (Heinrichsdorff et al., 1997). Note that, in addition to improving the crystal quality of the dots themselves, as discussed in Chapter 1, the crystal quality of the barrier layer should also improve because, if there is a large number of defects in the barrier, the injection efficiency of carriers into quantum dots decreases markedly. It was predicted that the discrete levels in quantum dots might severely disturb carrier relaxation from the excited states to the ground states (Bockelmann and Bastard, 1990; Benisty et al., 1991). This is the so-called phonon bottleneck effect. Whether this effect creates serious problems for quantum-dot laser operation is hotly debated. Some theoretical studies say that multiphonon relaxation enhances carrier relaxation (Inoshita and Sakaki, 1992; Heitz et al., 1996), while some experimental results indicate a carrier relaxation time of 10-loops (Mukai et al., 1996b and c). As described in Chapters 1 and 5, lasing characteristics are greatly influenced by carrier relaxation time. In particular, modulation characteristics, differential quantum efficiency, and maximum output power are severely limited by the photon bottleneck, erasing the advantages of quantum-dot lasers over conventional quantum-well lasers (Sugawara et al., 1997b). We need to clarify the effect of the phonon bottleneck. If it is serious, a new structure for accelerating carrier relaxation must be designed and fabricated to avoid the bottleneck. In the design of dot size as well, the phonon bottleneck should be taken into account. The above issues have been discussed from a physical point of view. From a practical point of view, emission wavelength from quantum dots becomes important. For example, in InAs-GaAs quantum dots, the emission wavelength is typically between 0.9 pm and 1.2 pm, for which unfortunately, it is not easy to find practical applications. For that reason, a new dot structure must be realized that emits at practical wavelengths such as 1.3 pm and 1.55 pm, which are widely used in optical communications.

IV. Key Technologies for the Next Era Quantum-dot structures grown in the SK mode have been a breakthrough in fabricating quantum-dot lasers. However, their performance is still unsatisfactory, and the problems discussed in the previous section must be overcome before we see any improvement. In this section, we will look at several emerging technologies that will be critical in the next era of optoelectronics.

270

HAJIME SHOJI

1. CLOSELY STACKED QUANTIJM-DOT LASERS

Multiply stacked quantum-dot structures increase optical confinement to the active layer. Unlike in-plane density, dot density can be easily controlled, and optical confinement can be more widely tuned only by changing the number of stacking layers. For this purpose, the first approach, shown in Fig. 6.5, was to grow multistacked dot structures with thick ( > 7-8 nm) barriers. In this case, the dot layers are not electronically coupled in the vertical direction due to the thick barriers. Each dot layer acts as an independent active layer. In this approach, the increased number of dot layers leads almost proportionally to enhancement of total modal gain, but modal gain from a single dot layer is unchanged. Closely stacked dot structures, that is, multistacked dot structures with thinner barriers, have been investigated by many (Solomon et al., 1996; Ledentsov et al., 1996; Nakata et al., 1997). Figure 6.20 shows (a) a cross-sectional TEM photograph of a InAs/GaAs closely stacked quantumdot structure and (b) the corresponding schematic illustration (Shoji et al., 1997~).By use of the thin GaAs barriers, the upper-layer dots were

FIG.6.20. Closely stacked quantum-dot structure. (a) a cross-sectional TEM photograph. (from Shoji et al., 1997c), and (b) the corresponding schematic.

6

SELF-ASSEMBLED QUANTUM DOTLASERS

271

self-aligned just on the lower-layer dots, forming a columnar structure, where the GaAs barriers separating the dot layers were as thin as 3 nm, and fivefold InAs quantum dot layers were grown (Nakata et al., 1997; Shoji et al., 1997~).Each dot layer of the fivefold structure was grown in the SK mode. It was found that the vertically aligned dots equivalently act as a single dot due to electronic coupling through tunneling in the vertical direction. Measurement of the exciton diamagnetic shift confirmed that the vertically aligned dot has an almost spherical quantum-confined potential and that the luminescence is due to three-dimensionally confined excitons (Sugawara et al., 1997a). The SK growth technique is described in Chapter 2. The closely stacked quantum-dot structure has features that make it attractive for laser application. The first one is the narrow emission spectrum obtained. Figure 6.21 shows a PL spectrum of the closely stacked quantum dots at 4.2K (solid curve) together with that of single-layer quantum dots without electronic coupling (dashed curve). The FWHM is as small as 25 meV in the closely stacked dot structure, which is much smaller than that of the conventional structures (Sugiyama et al., 1997; Shoji et al., 1997~).We can expect enhancement of optical gain due to small inhomogeneous broadening. One reason for the narrowed emission spectrum is the effect of size averaging during the stacking process. The upper-layer dots are grown on seed potentials formed by lower-layer dots, which results

1 '

.e I-

0.9

I

I

t

'I ' I

8 8

0.8

I

1.0

1.1

Wavelength

1.2

1.3

(vm)

FIG.6.21. PL spectrum of a closely stacked quantum dot at 4.2 K (solid curve). together with that of a conventional quantum dot without electronic coupling (dashed curve). The FWHM is as small as 25 meV in the closely stacked dot structure. (From Shoji et al., 1997c.)

272

HAJIME SHOJI

in size averaging in the lateral direction. The other reason is the larger volume of quantum dots. Because the effective vertical height becomes greater than that of single-layer dots, the emission wavelength becomes less sensitive to fluctuations in dot size. The red shift of the PL peak in Fig. 6.21 is further evidence of the larger volume of quantum dots due to the electronic coupling of stacked dot layers. The second attractive feature is enhancement of modal gain from a single dot layer. In the closely stacked structure, optical confinement to the active layer is enhanced by the number of stacked layers. Because the closely stacked dots act as a single dots, the modal gain obtained from a single dot layer becomes larger. Larger optical gain is expected in the closely stacked structure at an injection level that compares with that of the structure without vertical coupling. Effective dot height can be precisely controlled by changing the number of stacking cycles. Laser oscillation has been achieved by use of the closely stacked quantum-dot structure. Shoji et al. (1997~)reported pulsed lasing at room temperature and CW lasing at 50 K. The threshold current density at 50 K was 520 A/cm2. As shown in Fig. 6.22, lasing occurred at the second subband at 50 K. Bimberg et al. (1996) demonstrated a laser oscillation using a different closely stacked structure, where the thickness of GaAs barriers separating InAs dot layers was 6nm. Even for this rather thick barrier, a decrease in the FWHM of the quantum-dot emission down to

L = 900 pm

as-cleaved > c .u) c

0)

c

C

900

1000

1100

1200

Wavelength (nm) FIG. 6.22. EL and lasing spectra of closely-stacked quantum dot lasers for various injection currents at 50 K. Lasing occurs at the second state. (From Shoji et al., 1997c.)

6 SELF-ASSEMBLED QUANTUMDOTLASERS

273

30-50meV was observed. As for lasing, a threshold current density of 270 A/cm2 was achieved at 77 K, and improvement in the characteristic temperature was confirmed. The problem with this structure was its large nonradiative recombination rate, which led to the drastic increase in the threshold current at room temperature. This might be related not only to the growth conditions, which was not yet optimized, but also to the increased number of growth interfaces in the closely stacked structure. In the result shown in Fig. 6.22, the temperature dependence of the PL intensity showed that the peak intensity at room temperature was less than one-thousandth of that at 77K. In addition, the emission efficiency at 77 K was lower than that of conventional single-layer dots grown in the SK mode. These properties suggest that the closely stacked dots still contained many defects or nonradiative centers introduced during the growth, which degraded the optical properties of the quantum dots even at low temperatures. By improving the growth technique, we will see lower threshold current density operation at the ground state of quantum dots and improved gain properties due to the reduced fluctuation of size and composition in quantum-dot lasers with a highly uniform closely stacked structures.

2. COLUMNAR QUANTUM-DOT LASERS More recently, improved lasing characteristics have been demonstrated by Mukai et al. (1998a) using a columnar quantum-dot structure shown in a cross-sectional TEM image in Fig. 6.23. It is similar to the closely stacked structure with a columnar shape, but the GaAs barrier layers separating the InAs islands are much thinner. Because only three monolayers of GaAs were supplied on the InAs islands in each cycle of the stacking process, the existence of GaAs barriers was not clearly observed between the upper and lower islands. InAs islands produced during a cycle of InAs supply seemed to be contacting each other, forming equivalent single dots with strong electronic coupling in the vertical direction. The diameter and the height were about 15 nm, and the ground-state emission peak was observed at 1.17 pm. In the columnar quantum dots, the PL emission intensity was improved by three orders of magnitude compared with that of the closely stacked dots. Although the FWHM of the PL spectrum was about 40meV, which was larger than that of the closely stacked dots, the value was almost half that of ordinary quantum dots without electronic coupling. Figure 6.24 shows light output versus current characteristics of a 900-pm-long device with as-cleaved facets for various temperatures up to 70°C. The threshold current

274

HAJIMESHOJI

I

10 nm

I

FIG. 6.23. Cross-sectional T E M image of a columnar quantum dot.

was 31 mA at 2 5 T , which corresponds to the threshold current density of 500 A/cm2. This value was improved by two orders of magnitude compared with as-cleaved devices with closely stacked dot structures. As the temperature increased from 25°C to 70°C the threshold current increased up to 54 mA, which corresponds to the characteristic temperature of 81 K. Furthermore, a large output power without saturation was maintained up to high temperature, and the temperature dependence of the slope efficiency was considerably smaller, as shown in Fig. 6.24. In the low temperature range, the temperature characteristics were much better than those at room temperature. The threshold current density at 100 K was as low as 80 A/cm2, and the characteristic temperature defined between 100 K and 160 K was as high as 487 K. Lasing occurred at the second state at room temperature and at the ground state below 220 K. In devices with smaller mirror loss-for example through high-reflection coating applied to the facets- room-temperature lasing at the ground state with much lower threshold current, much better temperature characteristics. and greater output power is expected (Mukai et al., 1998b). Greater output power without any saturation and greatly reduced threshold current are attributed not only to improved emission efficiency but also to a possible enhancement of the carrier relaxation rate. Figure 6.25 shows the PL spectrum measured at 300 K together with that of the single-layer dots grown in the SK mode. An important point here is the position of the emission peak from the wetting layer. The peak wavelength was at around 1.0 pm, which was 0.1 pm longer than that of single layer dots (Shoji et al.,

6 SELF-ASSEMBLED QUANTUM DOTLASERS

275

F E

v

0

40

20

60

80

100

120

Current (mA) FIG. 6.24. Light output versus current characteristics of a columnar quantum-dot laser. showing the result of a 900-pm-long device with as-cleaved facets for various temperatures up to 70°C. The threshold current is 31 mA at 25"C, which corresponds to the threshold current density of 500 A/cm2. (Reprinted with permission from Mukai et al., 1998. :p 1998 IEEE.)

I ui .-c c

-

- 20 meV H

300K

I

Columnar dots

3

$ -

Wetting layer

v

%

a

Wetting layer

0.8

0.9

1.0 1.1 1.2 Wavelength (Vm)

1.3

1.4

FIG. 6.25. PL spectrum of columnar quantum dots at 300K together with that of a single-layer dot grown in the SK mode.

276

HAJIMESHOJI

1997a). The emission from the wetting layer at a longer wavelength was due to electronic coupling of thin wetting layers associated with extremely thin barrier layers separating the dot layers, which resulted in the formation of an effectively thick wetting layer, as in a quantum well. As discussed in Chapter 5, the lasing characteristics of quantum-dot lasers are subject to the carrier relaxation rate from the continuous level to the quantum confined states (Sugawara et al., 1997b). If the energy difference between the continuous state in the wetting layer and the discrete levels of quantum dots is smaller, the carrier relaxation rate should be enhanced by the assistance of the multiphonon effect (Inoshita and Sakaki, 1992) because the number of required multi-phonons is smaller for the shallow potential.

3. LONG-WAVELENGTH QUANTUM-DOT LASERS Quantum dots that emit in the long wavelength region such as 1.3 pm and 1.55pm are of great practical interest. They can be used in a variety of practical devices, such as optical communication and optical interconnects. Emission in the 1.3-pm region has been reported by several groups in InAs/GaAs or in InGaAs/GaAs quantum-dot structures on GaAs substrates (Mukai et al., 1994; Tackeuchi et al., 1995; Mirin et al., 1995). On InP substrates, emission in the 1.4- 1.7-pm spectral region has been demonstrated in InAs/InAlAs or InAsPnGaAsP quantum-dot structures (Fafard et al., 1996; Nishi et al., 1998). As described in Chapter 3, the ALS quantum-dot structure, which is self-organized by alternately supplying monolayers of InAs and GaAs in the atomic layer epitaxy (ALE) mode, holds promise for realizing long wavelength emission in quantum dots. ALS dots are In,,,Ga,,,As quantum dots typically with a diameter of 20 nm and a height of 10 nm, and they are surrounded by a 10-nm-thick Ino,,Ga,,,As barrier in the lateral direction and by a GaAs barrier in the growth direction (Mukai et al., 1994). The emission wavelength of the ALS dots was about 1.3 pm at room temperature, as shown in Fig. 6.26, and the wavelength can be controlled in a range of 1.1-1.5pm by changing the growth conditions, such as the number of cycles for alternate supply of InAs and GaAs monolayers and the composition of the InGaAs buffer layer on which quantum dots are grown (Ohtsuka et al., 1995). The FWHM of the ground-state PL peak was as narrow as 35 meV. These features are quite attractive for laser application. Using the ALS dots, quantum-dot lasers were fabricated by Shoji et al. (1995). One is schematically shown in Fig. 6.27. In the active layer, InGaAs quantum dots were self-organized by performing 12 cycles of (lnAs),/ (GaAs), short-period growth. Figure 6.28 shows the emission spectra for

6 SELF-ASSEMBLED QUANTUM DOTLASERS

271

Wavelength (vm) FIG. 6.26. PL spectrum of ALS quantum dots at 300K. The emission peak is at 1.3 jtm, and the FWHM is as small as 35meV. (From Shoji et al., 1995. Ci 1995 JEEE.)

various injection currents at 80 K. Although individual peaks were not as clear as those in the PL spectrum shown in Fig. 6.26, discrete quantum levels in the EL spectra were found under current injection as well as under photo-excitation (Mukai et al,, 1996b). Band filling with the current injection was clearly observed in this measurement. Several high-order levels appeared in the shorter wavelength region with increasing injected current because of a higher excitation level compared with the PL measurement. Laser oscillation was achieved at the threshold current of 1.1A under a Ino,Gao ,As Quantum dot

p+-GaAs contact (0.5 pm)

- - - - - - - - -'\

*n-In, 47Ga053Pclad (1 pm) n-GaAs buffer (0.5 pm)

\

*I1:

1

In, ,Ga, ,As barrier n-GaAs sub.

FIG. 6.27. Schematic of a n ALS quantum-dot laser. The active region consists of singlelayer ALS dots formed by twelve cycles of (InAs),/(GaAs), short-period growth. (From Shoji et a]., 1995. (3 199.5 IEEE).

278

HAJIME SHOJI

.-v)

1.1 A (lasing)

80 K

40

4-

C

3

.@ 30

m

Y

.220 v)

5

W

4-

C

- 10 0 800

1000 1200 Wavelength (nm)

1400

FIG. 6.28. Emission spectra for various injection currents at 80 K. Although the peaks are not as clear as in the PL spectrum shown in Fig. 6.26, discrete quantum levels in the EL spectra are found under current injection. Due to band filling with current injection, several high-order levels appear in the shorter wavelength region because of a higher excitation level compared with the PL measurement. Laser oscillation is achieved at the threshold current of 1.1 A under the pulsed condition. (From Shoji et al., 1995. 8 1995 IEEE.)

pulsed condition, and the corresponding current density was 815 A/cm2. The lasing wavelength was 91 1 nm. Considering the fact that the compositional wavelength of the 10-nmthick In,,,Ga,,,As barrier layer at 80 K was shorter than 850 nm, the laser oscillation was from a high-order sublevel of the quantum dots. Note that the wetting layer, which is formed in the growth of quantum dots in the SK mode is not observed in the ALS quantum-dot structure. To confirm that the obtained laser oscillation was really from a high-order sublevel of quantum dots, the diamagnetic shifts of the lasing peak were also carried out. A strong magnetic field up to 13T was applied perpendicular to the lasers, and the experimental results showed that the energy shift of the lasing peak was much smaller than that of the quantum well, whose compositional wavelength was close to that of the InGaAs barrier layer surrounding the quantum dots. This indicates that the obtained laser oscillation was from the quantum-confined level of the quantum dots. In this experiment, although lasing was observed up to 140 K, room-temperature operation was not achieved because the dot density was found to be quite low (a few percent areal coverage) and only a single-sheet dot layer was grown in the active layer. In addition, because of the limitation in growth equipment, the

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active layer was exposed to the air once before growing the p-type cladding layer, so that nonradiative centers might have been introduced. Considering the emission intensity at room temperature, as increase in the dot density and improved growth would lead to 1.3-pm operation and a lower threshold current at room temperature. Mirin et al. (1995) reported a sharp emission at 1.3 pm from In,.,Ga,,,As quantum dots grown in the SK mode. Although these dots were larger than the ALS quantum dots, the FWHM of the emission was as narrow as 28meV. These researchers also succeeded in laser oscillation at room temperature (Mirin et al., 1996). Because the lasing occurred at an excited state, the lasing wavelength was 1.2 pm. However, this is so far the longest lasing wavelength ever reported in self-assembled quantum-dot lasers grown on GaAs substrates. Nishi et al. (1998) reported a possibility of long wavelength quantum-dot lasers on InP substrates. InAs quantum dots were grown on (311)B InP substrates in the SK mode. Although lasing was still at low temperatures, the lasing wavelength of 1.2- 1.4 pm and the threshold current of 540 A/cmZ were demonstrated at 77 K. If room temperature lasing can be achieved, a much longer lasing wavelength is expected in this structure. Note that in the case of InAs dots on InP substrates the dot diameter is typically 30 nm or more (Fafard et al., 1996; Nishi et al., 1998), which is larger than that of InAs dots on GaAs substrates at present. This may originate from the small lattice mismatch between the lnAs and the substrate. Extracting the atomlike nature from quantum dots requires a smaller size.

4. QUANTUM-DOT VERTICAL-CAVITY SURFACE-EMITTING LASERS

Recent remarkable progress in the development of vertical cavity surface emitting lasers (VCSELs) has enabled the threshold current of semiconductor lasers to be drastically reduced. Ultra-low threshold currents below 100pA have been demonstrated in VCSELs in which strained quantum-well structures were used in the active regions (Huffaker et al., 1994; Hayashi et al., 1995). For much more advanced lasers, quantum-dot structures are an attractive approach. Ultimately low threshold current operation can be expected in quantum-dot VCSELs because of the enhanced gain characteristics of the quantum-dot’s active region and the reduction of active volume in the microcavity structure. Control of the electronic states and the photon modes in the microcavity structure is another interesting scheme to be studied from both practical and physical points of view (Baba et al., 1991). As described in Fig. 6.29, if the cavity mode is completely matched with the narrow gain bandwidth of quantum dots, a much lower threshold current

280

HAJIME SHOJI

E t -c Mirror

Quantum dots

Mirror

Optical field

Photon mode

nth

(n-1)th

3rd

2nd

Ground

Wavelength

FIG. 6.29. Photon modes and electronic states in a quantum-dot microcavity, where both the photon mode and the electronic state become discrete. Although lasing occurs only when the photon mode and the electronic state match, an extremely low threshold current is expected.

should be achieved because only a limited number of resonant modes can exist in the microcavity. Along with their merits, there is the potential that ideal quantum-dot VCSELs might work only at a fixed operating condition. The narrow gain bandwidth would make it difficult to match the gain peak with a resonant wavelength of the cavity. A difference in the temperature dependence between the gain peak and the resonant wavelength would also limit the operating temperature or the operating current. On the other hand, in the actual quantum-dot structure with large inhomogeneous broadening, a small active region would effectively reduce the inhomogeneous broadening, leading to the enhancement of gain properties. In this case, demonstrating quantum-dot VCSELs is rather easier. The first demonstration of quantum-dot VCSELs by current injection was reported by Saito et al. (1996). Figure 6.30 shows a schematic structure of the quantum-dot VCSEL. The active layer consisted of 10 periods of In,.,Ga,.,As quantum-dot layer grown in the SK mode and a 10-nm-thick Al,,,,Ga,,,,As barrier. The top and bottom mirrors consisted of 14.5-

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FIG. 6.30. Quantum-dot VCSEL with 10-period InGaAs dots in the active region. (Reprinted with permission from Saito et al., 1996. 6 1996, American Institute of Physics.)

period and 18-period AIAs/GaAs distributed Bragg reflectors (DBRs), respectively. The active layer was set at the center of the cavity, and the cavity size was adjusted to one wave. The resonant wavelength was around 960 nm. CW lasing at room temperature was achieved in the fabricated devices with 25-pm x 25-pm apertures, and the threshold current was 32 mA. From the measurement of emission spectra at various injection currents, the lasing was found to be associated with a higher-order transition in the quantum dots. It is well known that polarization of VCSELs on (100) substrates is not easily controlled because the optical gain is isotropic in the (100) plane. To control the polarization of VCSELs, Saito et al. (1997) proposed an interesting application of quantum dots. They used the structural anisotropy of the InGaAs quantum dot structure on (100) GaAs, rather than the change in the density of states, for polarization control of VCSELs. The shape of the grown dots was longer in the [ O i l ] direction on the (100) surface, which resulted in a polarization dependence of PL intensity along the [ O i l ] direction that was 1.37 times stronger than that along the orthogonal [011] direction. This anisotropy resulted in the asymmetric gain characteristics in the active layer plane of VCSELs. As shown in Fig. 6.31, in the VCSELs with the anisotropic quantum dots, lasing occurred only in the [ O i l ] polarization state, while the orthogonal [01 13 polarization was suppressed by 18 dB. Saito et al. also achieved ground-state pulsed lasing in these devices by increasing the periods of DBRs compared with their previous work (Saito et al., 1996). Sub-mA operation at room temperature has been achieved by Huffaker et al. (1997). They introduced a dielectric aperture by the use of selective oxidation of AlAs for three-dimensional optical confinement in the cavity

282

HAJIME SHOJI 1.6 Pulse 1.4 1.2

2 E 1.0 v 4-.

2

I

0.8

2

0

E 0.6 rn ._ 1

0.4 0.2

0 0

20

40

60

80

100 120

Current (mA)

FIG. 6.31. Light output versus current for two polarization states of a quantum-dot VCSEL at room temperature. The orthogonal polarization suppression ratio is 18 dB at 1-mW output power of a [OTl] emission. (Reprinted with permission from Saito et al., 1997. 0 1997 American Institute of Physics.)

(Dallesasse et al., 1990; Hayashi et al., 1995). The fabricated quantum-dot VCSEL with a 7-pm-square active region yielded a low threshold current of 560pA at room temperature under a pulsed condition, and the lasing was nearly from the ground states. A small aperture of less than 10pm associated with the index guiding effect resulted in the sub-mA operation of the quantum-dot VCSEL.

V.

Conclusion

As described in this chapter, self-assembled quantum-dot structures represent a breakthrough in quantum-dot lasers, which until that point had not been easily realized despite much effort since they were first proposed in 1982. Room-temperature operation through the introduction of self-assembled structures was a milestone in the history of quantum-dot laser research. Measurement of the fabricated quantum-dot lasers has given us much quantitative information. On the other hand, there is a big gap between ideal and actual performance. Many problems must be solved if the performance of quantum dots is to match what has been theoretically

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predicted. Although the self-assembled structure easily produces quantum dots, the growth process is based primarily on spontaneous mechanisms, and it is not yet fully controlled. We have to find a way to artificially control the uniformity, size, density, and position as well as the crystal quality of grown dots. In this sense, use of vertically coupled quantum-dots such as closely stacked and columnar quantum dots is of interest. In addition, we need to find an application for quantum-dot lasers. As described in Section IV, lasing wavelength becomes a key issue in finding practical uses for them. Realization of quantum-dot lasers that operate in practical wavelength ranges, such as 1.3 pm or 1.55 pm, would be advantageous. Exploitation of a new function for quantum dots would be also required.

Acknowledgments

The author would like to thank many colleagues, K. Mukai, Y. Nakata, N. Ohtsuka, Y. Sugiyama, T. Futatsugi, Dr. M. Sugawara, Dr. S. Yamazaki, Dr. N. Yokoyama, and Dr. H. Ishikawa who contributed to the work at Fujitsu Laboratories Ltd. The author also would like to thank K. Otsubo, Dr. H. Kuwatsuka, T. Fujii, Dr. S. Yamakoshi, Dr. K. Wakao, and Dr. H. Imai for their encouragement and fruitful discussion.

REFERENCES Ahopelto, J., Lipsanen, H., and Sopangen. M. (1995). Proc. 7th Con/: Indium Phosphide rind Relutrd Materiuls ( I P R M '9s) FB2-4, 19-20. Arakawa, Y., and Sakaki, H. (1982). Appl. Phys. Lett. 40, 939-941. Arakawa, Y., Sakaki, H., Nishioka, M., Okamoto, H., and Miura, N. (1983). Jpn. J . A p p l . Phy.~. 22, L804-L806. Arakawa, Y., and Yariv, A. (1986). IEEE J. Qimritton Electron. QE-22, 1887-1899. Asada, M., Miyamoto, Y., and Suematsu, Y. (1986). ZEEE J. Quantuni Electron. QE-22, 1915-1921. Baba, T., Hamano, T., Koyama, F., and Iga. K. (1991). IEEE J. Qumturn Electrori. QE-27. 1347 1 354. Benisty, H.. Sotomayor-Torres, C. M., and Weisbuch, C. (1991). Phys. ReG. B.44, 10945- 10948. Bimberg, D., Ledentsov, N. N., Grundniann, M., Kirstaedter, N., Schmidt, 0. G., Mao. M. H., Ustinov, V. M., Egorov, A. Y., Zhukov, A. E., Kop'ev, P. S., Alferov, 2. I., Ruvimov, S. S., Gosele, U., and Heydenreich, J. (1996). J p n . J . Appl. Phys. 35, 1311-1319. Bimberg, D., Kirstaedter, N., Ledentsov. N. N., Alferov, Z. I., Kop'ev, P. S., and Ustinov, V. M. ( I 997). IEEE J. Selected Topics in Qurintuni Electron. 3, 196-205. Bockelmann, U., and Bastard, G. (1990). Phys. Reu. B. 42, 8947-8953. Dallesasse, J. M., Holonyak, N.. Sugg. A. R., Richard, T. A., and El-Zein, N. (1990). Appl. Phys. Lett. 57. 2844-2846. -

284

HAJIMESHOJI

Fafard, S., Wasilewski, Z., McCaffrey, J., Raymond, S., and Charbonneau, S. (1996). Appl. Phys. Lett. 68, 991-993. Fukui, T., Ando, S., Tokura, Y., and Toriyama, T. (1991). Appl. Phys. Lett. 58, 2018-2020. Glaser, E. R., Benett, B. R., Shanabrook, B. V., and Magno, R. (1996). Appl. Phys. Lett. 68, 36 14-3616. Grundmann, M., Christen, J., Ledentsov, N. N., Bohrer, J., Bimberg, D., Ruvimov, S. S., Werner, P., Richter, U., Gosele, U., Heydenreich, J., Ustinov, V. M., Egorov, A. Y., Zhukov, A. E., Ustinov, V. M., and Alferov, Z. I. (1995). Phys. Reu. Lett. 74, 4043-4046. Grundmann, M., Ledentsov, N. N., Stier, 0..Bimberg, D., Ustinov, V. M., and Alferov, Z. I. (1996). Appl. Phys. Lett. 68, 979-981. Hatami, F., Ledentsov, N. N. Grundmann, M., Bohrer, J., Heinrichsdorff, F., Beer, M., Bimberg, D., Ruvimov, S. S., Werner, P., Gosele. U., Heydenreich, J., Ivanov, S. V., Meltser, B. Y., Kop’ev, P. S., and Alferov, Z. I. (1995). Appl. Phys. Lett. 67, 656-658. Hayashi, Y., Mukaihara, T., Hattori, N., Ohnoki, N., Matsutani, A,, Koyama, F., and Iga, K. (1995). Electron. Lett. 31, 560-561. Heinrichsdorf, F., Krost, A., Grundmann, M., Bimberg, D., Kosogov, A. O., and Werner, P. (1996). Appl. Phys. Lett. 68, 3284-3286. Heinrichsdorff,F., Mao, M. H., Kirstaedter, N., Krost, A,, Bimberg, D., Kosogov, A. O., and Werner, P. (1997). Appl. Phys. Letr. 71, 22-24. Heitz, R., Grundmann, M., Ledentsov, N. N.. Eckey, L., Veit, M., Bimberg, D., Ustinov, V. M., Egorov, A. Y., Zhukov, A. E.. Kop’ev, P. S., and Alferov, Z. I. (1996). Appl. Phys. Lett. 68, 361-363. Hiraydma, H., Matsunaga, K., Asada, M., and Suematsu, Y. (1994). Electron. Lett. 30, 142-143. Huffaker, D. L., Shin, J., and Depper, D. G. (1994). Electron. Lett. 30, 1946-1947. Huffaker, D. L., Baklenov, O., Graham, L. A,, Streetman, B. G., and Depper. D. G. (1997a). Appl. Phys. Lett. 70, 2356-2358. Huffaker, D. L., and Deppe, D. G. (1997b). .4ppl. Phys. Lett. 71, 1449-1451. Inoshita, T., and Sakaki, H. (1992). Phq’s. Ret:. B. 46, 7260-7263. Kamath, K., Battacharya, P., Sosnowski, T., and Phillips, J. (1996). Electron. Lett. 30, 1374-1375. Kamath, K., Phillips, J., Jiang, H., Singh, J.. and Bhattacharya, P. (1997a). Appl. Phys. Lett. 70, 2952-2953. Kamath, K., Klotzkin, D., and Bhattacharya, P. (1997b). Proc. 10th Ann. Meer. LEOS (LEOS ’97), ThY4. Kash, K., Scherer, A,, Worlock, J. M., Craighead, H. G., and Tarnargo, M. C. (1986). Appl. Phys. Lett. 49, 1043-1045. Kirstaedter, N., Ledentsov, N. N., Grundmann, M., Bimberg. D., Ustinov, V. M., Ruvimov, S. S., Maximov, M. V., Kop’ev, P. S., Alferov, Z. I., Richter, U., Werner, P., Gosele, U., and Heydenreich, J. (1994). Electron. Lett. 30, 1416-1417. Kirstaedter, N., Schmid, 0. G., Ledentsov, N. N., Bimberg, D., Ustinov, V. M., Egorov, A. Y., Zhukov, A. E. Maximov, M. V., Kop’ev. P. S., Alferov, Z. I. (1996). Appl. Phys. Lett. 69, 1226-1228. Ledentsov, N. N., Bohrer, H., Bimberg, D., Kochnev, I. V., Maximov, M. V., Kop’ev, P. S., Alferov, Z. I., Kosogov. A. O., Ruvimov. S. S., Werner, P., and Gosele, U. (1996). Appl. Phys. Lett. 69, 1095-1097. Leonard, D., Kishnamurthy, M., Reaves. C. M.. Denbars, S. P., and Petroff, P. M. (1993). Appl. Phys. Lett. 63, 3203-3205. Mao, M. H., Heinrichsdorff, F., Kirstaedter, N., Krost, A., Ledentsov, N. N., Bimberg, D., Ustinov, V. M., Egorov, A. Y., Zhukov, A. E., Kop’ev, P. S., and Alfreov, Z. I. (1997). Proc. 10th Ann. Meet. LEOS (LEOS ’97).ThY5.

6 SELF-ASSEMBLED QUANTUMDOTLASERS

285

Mirin, R. P., Ibbetson, J. P., Nishi, K., Gossard, A. C., and Bowers, J. E. (1995). Appl. Phps. Lett. 67, 3795-3191. Mirin, R. P., Gossard, A,, and Bowers, J. E. (1996). Electron. Lett. 32, 1732-1733. Moritz, A., Wirth, R., Hangleiter, A., Kurtenbach, A,, and Ebert, K. (1996). Appl. Phys. L e t f . 69, 212-214. Mukai, K., Ohtsuka, N., Sugawara. M., and Yamazaki, S. (1994). Jpn. J . Appl. Phys. 33, L1710-Ll712. Mukai, K., Ohtsuka, N., and Sugawara, M. (1996a). Jpn. J . Appl. Phys. 35, L262-L265. Mukai, K., Ohtsuka, N., Shoji, H., and Sugawara, M. (1996b). Appl. Phys. Lett. 68, 3013-3015. Mukai, K., Ohtsuka, N., Shoji, H., and Sugawara, M. (1996~).Phys. Rev. B. 54, R5243-R5246. Mukai, K.. Shoji, H., and Sugawara. M. (1998a). Proc. lO!h In!. Conj: Indium Phosphide and Related Muteriuls ( I P R M '98), pap. WBI-4. Mukai, K. (1988b) (unpublished). Nakata, Y., Sugiyama, Y., Futatsugi, T., Yokoyama, N. (1997). J . Cryst. Growth 175/176, 71 3-719. Nishi, K., Yamada, M., Anan, T., Gomyo. A.. and Sugou, S. (1998). Ext. Ahs. Jupunese Sociery of Applied Physics, 3 (in Japanese). Notzel, R., Temmyo, J., Kamada, H., Furuta, T., and Tamamura, T. (1994). Appl. Phys. Lett. 65, 457-459. Ohtsuka, N., and Mukai, K. (1995). Proc. 7th Int. Conj: Indium Phosphide und Related Muterials ( I P R M '95), WP59. Oshinowo, J., Nishioka, M., Ishida, S., and Arakawa, Y. (1994). Appl. Phys. Lett. 65, 1421-1423. Pan, D., Zeng, Y. P., Wu, J., Wang, H. M., Chang, C. H., Li, J. M., and Kong, M. Y. (1997). Appl. Phys. Lett. 70, 2440-2442. Saito, H., Nishi, K., Ogura, I., Sugou, S.. and Sugimoto, Y. (1996). Appl. Phys. Lett. 69, 3 140-3 142. Saito, H., Nishi, K., Sugou, S., and Sugimoto, Y. (1997). Appi. Phjs. Lett. 71, 590-592. Schmidt, 0. G., Kirstaedter, N., Ledentsov, N. N., Mao, M. H., Bimberg, D., Ustinov. V. M.. Egorov, A. Y., Zhukov, A. E., Maximov, M. V., Kop'ev, P. S., and Alferov, Z. I. (1996). Electron. Lett. 32, 1302-1 304. Shoji, H., Mukai, K., Ohtsuka, N.. Sugawara, M., Uchida, T., and Ishikawa, H. (1995). IEEE Photon. Technol. Lett. 7, 1385- 1387. Shoji, H., Nakata, Y., Mukai, K., Sugiyama, Y., Sugawara. M., Yokoyama, N., and Ishikawa. H . (1996a). Jpn. J. Appl. Phys. 35, L903-L905. Shoji, H., Nakata, Y., Mukai, K., Sugiyama, Y., Sugawara, M., Yokoyama. N.. and Ishikawa, H. (1996b). Electron. Lett. 32, 2023-2024. Shoji, H., Nakata, Y., Mukai, K., Sugiyama, Y., Sugawara, M., Yokoyama, N., and Ishikawa, H. (1997a). IEEE J. Selected Topics in Quantuni Electron. 3, 188-195. Shoji, H., Nakata, Y., Mukai, K., Sugiyama. Y., Sugawara, M., Yokoyama. N., and Ishikawa, H. (1997b). Appl. Phys. Lett. 71, 193-195. Shoji, H., Nakata, Y., Mukai, K., Sugiyama, Y., Sugawara, M., Yokoyama, N., and Ishikawa, H. (1997~).Proc. 2nd ConJ Optoelutronics rind Communicution (OECC' '97). 9C2-5. Shoji, H.. Nakata, Y., Mukai, K., Sugiyama, Y., Sugawara, M., Yokoyama. N., and Ishikawa, H. (1997d). Proc. 10th Ann. MCCI.L E O S ( L E O S '97), pap. TbY2. Soda, H., Iga, K., Kitamura, C., and Suematsu, Y. (1979). Jprz. J . Appl. Phys. 18, 2329-2330. Solomon, G. S., Trezza, J. A., Marshall, A. F., and Harris, J. S. (1996). Phys. Rer. Lett. 76, 952-955. Solomon, G. S., Larson, M. C., and Harris, J. S. (1996). Appl. Phys. Lett. 69, 1897-1899. Sugawara, M., Nakata, Y., Mukai, K., and Shoji, H. (1997a). Phys. Rev. B. 55, 13155-13160. Sugawara, M., Mukai, K., and Shoji, H. (1997b). Appl. Phys. Lett. 71, 2791-2793.

286

HAJIMESHOJI

Sugiyama, Y., Nakata, Y., Imamura, K.. Muto, S., and Yokoyama, N. (1996). Jpn. J . Appl. Phys. 35, 1320-1324. Sugiyama, Y., Nakata, Y., Futatsugi, T., Sugawara, M.. Awano, Y., and Yokoyama, N. (1997). Jpn. J . A p p l . Phys. 36, L158-LI61. Tackeuchi, A., Nakata, Y., Muto, S.. Sugiyama, Y., Inata, T., and Yokoyama, N. (1995). Jpn. J . Appl. Phys. 34, L405-L407. Temmyo, J. Kuramochi, E., Sugo, M., Nishiya, T., Kamada, H., Notzel, R., and Tamamura, T. (1994). Proc. 14th Int. Semiconductor Luser Cot$ (ISLC '94). PD3. Uskov, A. Y., McInerney, J., Adler, F., Schweizer. H., and Pilkuhn, M. H. (1998). Appl. Phys. Leu. 72, 58-60. Ustinov, V. M., Egorov, A. Y., Kovsh. A. R.. Zhukov, A. E., Maximov, M. V., Tsatsulnikov, A. F., Gordeev, N. Y., Zaitsev, S. V.,Shernyakov, Y. M., Bert, N. A,, Kop'ev, P. S., Alferov, Z. I., Ledentsov, N. N., Bohrer, J., Bimberg, D., Kosogov, A. O., Werner, P., and Gosele, U. (1996). Proc. 9th Int. Cortf: MBE. Xie, Q., Kalburger, A., Chen, A., and Madhukar, A. (1997). J . Cryst. Growth, 175/176,689-695.

SEMICONDUCTORS AND SEMIMETALS . VOL. 60

CHAPTER 7

Applications of Quantum Dot to Optical Devices Hiroshi Ishikawa ELECTRON DFVICES A N D MATERIALS LARS FUJlTSU LABORATORIES LTU

.

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ATSUGI KANAC~AWA JAPAN

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. PROPERTIES OF QUANTUM DOTS. . . . . . . . . . . . . . . . . . . . 1 . The Quantum Dot us a T M V - L ~ WS.ystern I . . . . . . . . . . . . . . . 2 . Attractive Features of Quantum Dots.for Device Application . . . . . . . 111. QUANTUM DOTSFOR VERYHIGHSPEEDLIGHT MODULATION . . . . . . . . 1. The Need.for High-speed. Low- Wavelength-Chirp Light Sources . . . . . 2 . Direct Modulation of Quantuni-Dot Lasers . . . . . . . . . . . . . . 3 . The Quuntuin-Dot Intensity Modulator . . . . . . . . . . . . . . . .

IV . QUANTUM DOTSAS A NONLINEAR MEDIUM. . . . . . . . . . . . . . . . 1. The Needfor Large Nonlinearity with u Large Bandwidth . . . . . . . . 2. Ana!ysis of zl3’ . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. PERSISTENT HOLE BURNINGMEMORY. . . . . . . . . . . . . . . . . . 1 . Persistent Spectrul Hole Burning Memory Using Quantum Dots . . . . . 2 . E.xperiinentu1 Results . . . . . . . . . . . . . . . . . . . . . . . . 3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . SUMMARY A N D PERSPECTIVES ON QUANTUM-DOT OPTICAL DEVICES . . . . . 1. Trends in Optoelectronics . . . . . . . . . . . . . . . . . . . . . . 2 . Usesfor Quantum Dot Opticul Devices . . . . . . . . . . . . . . . . Acknowledyment . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I

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287 288 288 294 295 295 298 302 303 303 306 311 314 314 316 319 319 320 321 321 321

Introduction

The past several years have seen remarkable progress in the fabrication of quantum dots and quantum-dot lasers. Thanks to new self-organized growth technologies for example. as described in Chapter 6. ground-state CW lasing can now be achieved at room temperature (Kirstaedter et al., 1994; Shoji et al., 1996). Although the performance of quantum-dot lasers is as yet far from ideal. recent gains have enlarged our expectation of applying 287 Copyright ( 1999 by Academic Press All rights of reproduction in any form recerved ISBN 0-1?-752169-0 ISSN 0080-8784 99 $3000

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quantum dots to various optical devices. One of the reasons that we are so interested in quantum dots is that the performance and functions of optical devices based on current technologies, such as the quantum well or the strained quantum well, have reached their limit. In this respect, quantum dots are highly attractive because of their unique three-dimensional carrier confinement which may provide us new optical properties. At present, application research is concentrating mainly on quantum-dot laser, and even here there is much to be done. We do not know what is ultimately achievable with quantum-dot lasers because our understanding of the carrier dynamics in quantum dots is incomplete. Nevertheless, we can start experimental and theoretical studies to explore the advantages of quantum dots for device applications. In this chapter, I first review some of the attractive properties of quantum dots and then discuss several feasible applications. In Section 11, I look at the quantum dot as a two-level system, and summarize some of its advantages. In Section 111, I discuss the use of quantum dots for high-speed modulation of light for long-distance high-bit-rate communication systems. I review the problems with conventional technologies and then discuss the direct modulation of quantum-dot lasers and the feasibility of an external modulator using quantum dots. In Section IV, I discuss the use of quantum dots as a nonlinear medium. Realization of large nonlinear susceptibility using quantum structures has long been a dream. To see if such a dream is feasible, I perform a trial analysis of the third-order nonlinear susceptibility of quantum dots. In Section V, I discuss the use of quantum dots for high-density optical memories using persistent spectral hole burning. In Section VI, I summarize and discuss the roles of quantum-dot-based devices in the future of optoelectronics. Because of our so-far limited knowledge of the quantum dots, my discussions in this chapter are not concrete. I simplify my analysis to show the feasibility of quantum dots. From an engineering point of view, such an approach is sometimes more efficient that waiting for the establishment of a full-blown and rigid theory and complete experimental evidence.

11. Properties of Quantum Dots

1. THE QUANTUM DOTAS

A

TWO-LEVEL SYSTEM

What makes quantum dots attractive is that their carriers are strongly confined three-dimensionally in a very narrow region; this gives discrete energy levels. There are also many unresolved problems with quantum dots:

7 APPLICATIONS OF QUANTUM DOTTO OPTICAL DEVICES

289

how the carriers relax to the ground state (Bockelmann and Bastard, 1990; Bockelmann, 1993; Efros et al., 1995), the dephasing time of the ground-state wave function, the effect of the Coulomb interaction on the emission spectra (Hu et al., 1990), and so on. However, here I put aside such problems and focus on some of the fundamental properties of quantum dot, assuming it to be a simple discrete two-level system. Most of these properties can be illustrated by this assumption.

Linear Gain Following a conventional density-matrix analysis, linear gain can be calculated from the linear susceptibility, fl), of the two-level system as

where c is the light velocity; y~is the refractive index; R is the crystal volume; n is the dot number; e is the electron charge; E, is the vacuum dielectric constant; Y : and 4':' are the wave function (including the Bloch part) of the ground-state electron and hole, respectively; f;. and f , are the Fermi-Dirac distribution function; EL and E:, are the ground-state energy of the electron and hole, respectively; and y,, is the inverse of the dephasing time. The suffix i denotes the ith dot, and the summation is for all dots in a volume, R. The square of the matrix element in Eq. (7.1) can be expressed by a matrix element for bulk (Kane, 1957) as

where, $: and Ic/i are the envelope wave functions of the ground state; m, is the electron mass, mr6 is the electron effective mass at r6;E , is the band gap of the quantum-dot material; and A is the spin-orbit splitting energy. From Eq. (7.1) we see that, at resonance, where Ek - El = ho,the linear gain is inversely proportional to hy,,, which is a broadening determined by the dephasing time. We obtain a large gain for a small broadening of Wy,,,, which is highly attractive for semiconductor lasers because a high gain results in a very low threshold current. What is interesting about the linear gain is the refractive index dispersion. Figure 7.l(a) shows schematically the refractive index dispersion of a quantum well. Refractive index, 17, is governed by dispersions due to various oscillators (Adachi, 1982). Among them, the carrier-density-dependent re-

290

HIROSHIISHIKAWA High carrier density

Gain, change

.. .. ,* * .

Quantum well

Quantum dot

(a)

(b)

FIG. 7.1. Schematic of gain (dotted line) and the refractive index (solid line) spectra for (a) a quantum well, and (b) a quantum dot.

fractive index change results from the change in the optical gain spectrum caused by band filling, band-gap renormalization, and the effect of plasma resonance (see, for example, Bennett et al., 1990). Any change in the gain spectrum changes the refractive index through the Kramers-Kronig relation. In a typical quantum well with a high-density carrier injection, an increase in carrier density causes a reduction in the refractive index a t the wavelength of gain region. This reduction, Aq, amounts to -4 x 10-20n, where n is the carrier density per cubic centimeter (Lee et al., 1986). Thus, a change of 5 x 10'7cm-3 in carrier density gives a refractive index change of about 0.02. There is no plasma dispersion associated with carriers in the quantum dots. Moreover, the gain is symmetric around the resonant wavelength. Figure 4.l(b) shows schematically the gain and refractive index dispersion of a quantum dot. If the shift of the ground-state transition wavelength for a change in carrier number is very small, we obtain an almost zero index change for a change in carrier number at the resonant wavelength. In the actual case, a red shift of the resonant wavelength may take place when a carrier is added to the ground state to form a bi-exciton (Hu et al., 1990), and the addition of carriers to upper sublevels may also cause some change in the resonant wavelength through the Coulomb interaction. These shifts are yet to be analyzed quantitatively but they are presumed to be within a few meV. In actual quantum dots there is an inhomogeneous broadening, to be described later, which cancels out the change in refractive index near the gain peak. If we can reduce the plasma dispersion effect of carriers at the

291

7 APPLICATIONS OF QUANTUM DOTTO OPTICAL DEVICES

barrier layer and the wetting layer, we might obtain an almost zero refractive index change at the gain peak wavelength for the change in carrier number in the quantum dot. As will be discussed in Section 111, there is a possibility of obtaining no wavelength chirp under high-speed modulation in quantum-dot lasers. Third Order Nonlinear Susceptibility Because the structure of unbiased quantum dots is an inversion symmetry, we can expect them to have odd-order nonlinearity. The third-order nonlinear susceptibilities of the two-level system can be calculated using the density-matrix analysis as f3'=

4e4

--z E&

=

<

I y:l Iy I)I4( L - .t.) (hw,- EL + EL $- ihycC)(hwp - ho,+ ihy,,)(hw - Ei J"

+ E:;

-

ihy,,)

where we assume a four-wave mixing process and only the resonant terms are shown. The x’~’is doubled taking account of the spin. In Eq. (7.2), w,, ws, and w, are angular frequencies of the pump wave, signal wave, and conjugate wave, respectively. ycc and y,,,.are the dephasing rate of each level. In the degenerate case, that is, Q,, = oiS,this reduces to f3'(wp=

w s )=

We can expect a large

f3)

for the smaller dephasing rate y,,,, as in the case

292

HIROSHIISHIKAWA

of linear gain. Of course, this also holds for the nondegenerate case. The large x'j) encourages us to use quantum dots as a nonlinear medium for various device applications. The response speed, or a bandwidth of ~ ( j ) , depends on the dephasing rate yes. The bandwidth reduces for small y,,, as can be read from Eq. (7.4).

Rate Equation for Photons We can derive the rate equation for photons using x"’ and f 3 ' under the degenerate on-resonant case, that is, w = cop = o,= (Ef - EI)/A. We assume a single polarization optical field with an angular frequency of w. We start from the equation for the optical electric field E (Kuwatsuka et al., 1997) as

where r is the optical confinement factor to the dot region, p O is the magnetic permeability and factor 3/4 is the degeneracy factor of f3). Substituting dz to be the product of the group velocity ug and d t , transforming electric field to photon density, s, and assuming that all dots have the same size and composition (then omitting the suffix i), Eq. (7.5) reduces to, dS

-=

t.,T(G

dt

S ES)NS - 'ph

where, the term of photon lifetime z,, i

-

is added assuming a laser cavity and

n

(7.7)

(7.9)

N represents the inversion carrier density in the ground state of quantum dots, G is the linear gain, and 2 is the gain compression factor, which corresponds to the one often used to express the gain saturation in semiconductor lasers (Nagarajian et al., 1992). The gain compression factor is larger for small y,,. The gain saturation takes place at high optical power for a large E.

7 APPLICATIONS OF QUANTUM DOTTO OPTICAL DEVICES

293

Mod$cation for Actual Quantum Dots

Inhomogeneous broadening in quantum dots is due to the nonuniformity of dot size and composition. To take account of this inhomogeneous broadening, we may carry out the summation over the index, i; giving a different size and composition for each dot in Eqs. (7.1) and (7.3). Here, to show the effect of this fluctuation, we assume that the inhomogeneity gives the Gaussian distribution of the ground-state transition-angular frequency, w,,, whereas other parameters are the same. We also assume a Gaussian distribution as (7.10)

where 2Aw is the eC1 width of the inhomogeneous broadening. The corresponding full width at half-maximum (FWHM) is 1 . 6 7 8 ~Quantum . dots available with present technology give a broadening of 25-80 meV (Mukai et al., 1996a and b; Nakata et al., 1997), which is quite large when compared with the homogeneous broadening try,,, which may be on the order of a few meV or below. In this case, we may presume the homogeneous Lorenzian broadening function in Eq. (7.1) to be a 6 function. Thus, the peak gain is

(7.11) The linear gain is inversely proportional to the inhomogeneous broadening. To obtain large linear optical gain, or a large f 3 ) , we must reduce the broadening. In Eqs. (7.1) and (7.2), we did not take into account the carrier energy relaxation process within the dots, which is not well understood quantitatively. If the relaxation process can be properly incorporated, making the systems a three-level one, these equations can be modified. Recent experimental and theoretical studies reveal that the carrier relaxation to the ground state is 10- 100 ps, which is fairly slow when compared with that of the quantum well (Mukai et al., 1995; Mukai et al., 1996b; Uskov et al., 1997; Uskov et al., 1998). The slow carrier energy relaxation time gives an additional term in f 3 ) to increase the gain compression factor. This results in the gain saturation in the quantum-dot lasers at high optical output power.

294

HIROSHI ISHIKAWA

2. ATTRACTIVEFEATURES OF QUANTUM DOTSFOR DEVICE APPLICATION On the basis of their basic properties, and from the two-level model just described, the attractive features of quantum dots for device application can be enumerated as follows:

1. Their discrete energy levels may open up new applications for quantum dots. In the quantum well, light with an energy greater than the bandedge energy is absorbed. In quantum dots, there are transparent regions between the discrete levels when the inhomogeneous broadening is not large. In Section 111, we discuss an optical intensity modulator that makes use of this feature. 2. There is a possibility of obtaining a very small refractive index change at the resonant wavelength for the change in carrier number in the quantum dot. This will give a very small wavelength chirp under high-speed direct modulation in quantum-dot lasers, as will be discussed in Section 111. 3. Discrete energy levels also affect carrier dynamics. When the LOphonon energy does not coincide with the level separation, carrier energy relaxation is slow, causing the so-called phonon bottleneck (Bockelmann and Bastard, 1990). However, the slow rate of carrier energy relaxation means the slow excitation rate of carriers from the dot. This will be advantageous in an infrared photodetector using the subband transition because it gives a small dark current. Readers interested in the quantum-dot infrared photodetector should refer to Ryzhii (1996). 4. Large x'l) and f 3 ) will be obtained in quantum dots if we can reduce the inhomogeneous broadening. A large x(l) gives a large linear gain, which means that we can expect very low threshold current lasers. A large x(3) will enable us to realize optical nonlinear devices. The response speed of fl) and x’~’depends largely on the dephasing rate, y,,, which is determined by the carrier dynamics. The quantum dots' unique carrier dynamics may give a different dephasing rate from that of conventional quantum wells. Section IV discusses x(3’ together with carrier dynamics in optical nonlinear devices. 5. The surface density of a quantum dot grown in the typical SK mode is about 10"/cm2. If we have both large dot density and large nonuniformity, leading to many different transition energy levels (equivalent to large inhomogeneous broadening), we may be able to use this as a huge-capacity optical memory. The application of quantum dots in Tb/cm2 memory is discussed in Section VI.

7

APPLICATIONS OF QUANTUM DOTTO OPTICAL DEVICES

295

111. Quantum Dots for Very High Speed Light Modulation

1. THENEEDFOR HIGH-SPEED, LOW-WAVELENGTH-CHIRP LIGHTSOURCES Very high speed light modulation for optical communication systems has been one of the major targets of semiconductor laser research. For example the frequency response of Fabry-Perot lasers has reached to 25GHz (Bowers et al., 1986), and distributed feedback (DFB) lasers can perform single-spectrum lasing for long-haul transmission systems (see, for example, Ishikawa et al., 1987). However, direct modulation in long-haul optical communication systems is limited to 2.5 Gb/s, not because of the limited modulation bandwidth but because of the wavelength chirping under direct modulation. The chirping of a single-mode DFB laser limits the transmission distance in high-bit-rate systems because of the chromatic dispersion of optical fiber. The chromatic dispersion is typically - 17 ps/km/ nm in nondispersion shifted fiber at the wavelength of 1.55pm, where the fiber loss is the smallest. The goal is to realize high-speed modulation of laser light with the smallest, or at least a controlled, wavelength chirp. Here, I first review the problems in direct modulation of semiconductor lasers and those in the external modulation scheme based on present technologies. I then discuss two possibilities; the chirpless high-speed direct modulation of quantum-dot lasers and the use of quantum dots as an external optical intensity modulator with controlled wavelength chirping.

The Direct Modulation Scheme Wavelength chirping under direct modulation of semiconductor lasers is due to a change in the refractive index caused by a change in carrier concentration. It can be evaluated by (Koch and Linke, 1986; Wiesenfeld et al., 1987) (7.12) where P is the light power, r is the optical confinement factor, 0 is the gain compression factor (as discussed in Eq. (7.6)), V,,, is the active layer volume, yld is the differential quantum efficiency, and hv is the photon energy. The parameter u is (Henry, 1982) (7.13)

296

HIROSHIISHIKAWA

which is proportional to the ratio of the refractive index change to the gain change for the change in carrier concentration N . The first term in Eq. (7.12) gives the change related to the time change of the optical power originating from the phase modulation of light. The second term gives the offset wavelength shift, which depends on the optical power. The value of tl in multiple quantum well (MQW) lasers is around 3 to 4 (Green et al., 1987). In semiconductor lasers under very fast modulation, the second term is mostly responsible for the chirp (Wiesenfeld et al., 1987). Its coefficient depends largely on the device structure. Wavelength chirping is often evaluated by the time-averaged spectral width. In quantum-well DFB lasers, a broadened spectral width due to chirping of 0.28 nm (-20 dB full width) was reported under 2 Gb/s NRZ (non-return zero) modulation (Kakimoto et al., 1990), and one of 0.34nm was reported under 10Gb/s NRZ modulation (Uomi et al., 1991). The chirp at 10Gb/s modulation is large enough to limit the transmission distance in 10Gb/s systems. For a fiber dispersion of - 17 ps/km/nm, 0.34-nm chirp causes a pulse broadening of 578ps for a 100-km transmission. This value exceeds the time slot of 100ps for 10 Gb/s NRZ modulation.

Review of the External Modulation Scheme External modulation schemes have been studied extensively as a way to overcome the chirping problem. For example, the LiNbO, Mach-Zendertype modulator gives, in principle, zero-frequency chirp (Seino, 1996). It is used for lO-Gb/s long-distance systems. Semiconductor intensity modulators also are attractive because of their small size and easy monolithic integration with a DFB laser. They use either the Franz-Keldysh effect in the bulk absorption layer (Soda et al., 1988) or a Stark shift of the exciton peak wavelength in the MQW (Devaux et al., 1992) under an electric field application. There is still a wavelength chirp, due to phase modulation in the intensity modulator, however (Koyama and Iga, 1985), which is characterized by the tl value defined as (7.14) where the derivatives are by the applied voltage, I/: The wavelength chirp for the external modulation is given by substituting this a value in Eq. (7.12) and eliminating the second term, which is specific to semiconductor lasers. Figure 7.2(a) illustrates how the tl parameter is determined in the MQW modulator. As shown in the figure, application of the electric field causes a

7 APPLICATIONS OF QUANTUM DOTTO OPTICAL DEVICES

A

Exciton peak

297

A

Signal

a.

lv&o>a.]+o v=o (a) Normal operation

v = v, (]I

>0

v=o

+a.

a.

v = v,

(b) Prebiased operation

aC 0

FIG. 7.2. Parameter in a quantum well under bias-voltage application. In case (a) the usual operating condition of an MQW modulator i( parameter, defined as i(= -2k(?q/dI/)/dg/dI/; is positive. Case (b) is the prebiased operating condition. We obtain a negative r value.

shift of the exciton absorption peak. In association with this peak shift, the refractive index dispersion also shifts, as shown. The refractive index change is positive, and the gain change is negative; thus, the CI parameter is positive. This refractive index change gives the blue chirp at the turn-on and the red chirp at the turn-off, as can be read from Eq. (7.12). The blue-chirped leading edge of the light pulse is faster than the red-chirped pulse end because of a fiber dispersion. Therefore, this chirping widens the pulse width when transmitted through an optical fiber. In the MQW modulator, however, the CI value is typically 0.2- 1.4 (Kataoka et al., 1994), which is smaller than in the MQW laser. In addition, there is no second term of Eq. (7.12). Therefore, the wavelength chirp in external modulators is smaller than that of the direct modulation of semiconductor lasers. At lO-Gb/s NRZ modulation, a wavelength chirp of 0.15 nm (-20 dB full width) was reported, but this limits the transmission distnce to about 25 km (Haisch et al., 1994). If we can get a negative CI parameter, we can get pulse compression in the transmission system. When the chirp is controlled to give a narrow pulse width for a given transmission

298

HIROSHIISHIKAWA

distance, the signal-to-noise ratio improves. The negative a parameter can be realized in the MQW modulator by prebiasing the modulator or by setting the signal wavelength close to the exciton peak, as shown in Fig. 7.2(b). Recently this was done in the modulator-integrated DFB laser, and transmission over 100 km at 10 Gb/s was achieved without any penalty in the error rate (Morito et al., 1996). There is a disadvantage to this method, however. Because of the prebiasing, there is an absorption at light-on state, which causes the extinction ratio to deteriorate when compared with the unbiased operation. It is very difficult to get an extinction ratio larger than 15 dB. If we could make a modulator with a zero or negative a value without such loss, the performance of the semiconductor-based intensity modulator for high-bit-rate long-haul transmission systems would be much improved.

2. DIRECT MODULATION OF QUANTUM-DOT LASERS

Benefits of Quuntum-Dot Lasers The most attractive feature of quantum-dot lasers is that they make possible a very small refractive index change at the resonant wavelength, as explained in Section 11. This is something that conventional MQW lasers cannot do. If we could ignore the plasma effect of the carriers in the barrier layer or in the wetting layer, we might obtain an almost zero a value, which would lead to no wavelength chirp under high-speed direct modulation. The problem is how fast we can modulate quantum-dot lasers, which relates to the yet unresolved problem of carrier dynamics in quantum dots- the phonon bottleneck (Bockelmann and Bastard, 1990).

Modulation Response and Carrier Dynamics The modulation response of the quantum-dot lasers can be understood through the well-developed modulation theory of quantum-well lasers (Tucker, 1985; Koch and Linke, 1986; Olshanski et al., 1987; Nagarajian, 1992). The intrinsic maximum modulation bandwidth, which is the relaxation oscillation frequency, can be given by 1

(7.15)

where z p is the photon lifetime, E is the gain compression factor, vg is the

7 APPLICATIONS OF QUANTUM DOTTO OPTICAL DEVICES

299

group velocity of light, and x represents the ratio of carriers in the upper subband and the barrier to those in the ground state (Nagarajian et al., 1992; Ishikawa and Suemune, 1994). At first sight, one benefit of the quantum-dot laser is that the differential gain, G’ can be made very large if we can reduce the inhomogeneous broadening, because G’ is inversely proportional to the inhomogeneous broadening width. This gives a high f,,,. However, factors specific to quantum-dot lasers prevent a large fmax: factor and the gain compression factor E. Recent experimental and theoretical studies suggest a large carrier relaxation time to the ground state in quantum dots of 10-1OOps (Mukai et al., 1995; Mukai et al., 1996b; Uskov et al., 1997; Uskov et al., 1998). This is much slower than that in the MQW laser, which is around 10- 100 fs (Seki and Yokoyama, 1994). As the factor x is the ratio of total carriers to those in the ground state, it becomes large because of the slow energy relaxation. This slow energy relaxation also causes a large value of E resulting in gain saturation at high power levels. There is a possibility that the product of Zx will be a large value which would cancel out the large differential gain. There also is a frequency rolloff associated with carrier transport (Nagarajian, 1992). In SK-mode quantum dots, carriers are injected into the barrier layer and then diffuse to a quantum dot and relax to the ground state. These carrier transport time constants give the rolloff in modulation response, S(o,), given by (7.16) where omis the modulation angular frequency, zd is the diffusion time, and T ~ is the , carrier ~ ~ energy ~ relaxation time. If we assume a diffusion distance of 50 nm and that a diffusion constant of 7 cm2/s, zd is 1.8 ps, and if we assume a Zrefax of 20 ps, Eq. (7.16) gives a rolloff of -2.1 dB at a modulation frequency of 10GHz. This roleoff is mainly due to the energy relaxation time. Experimental Reports Kamath et al. (1997) prepared two types of In,,,Ga,.,As/GaAs quantumdot laser. One type is the laser with GaAs SCH layers. They made lasers with different stack of dot layer, The other is a tunnel-injection type in which an Al,.,Ga,.,As thin tunneling barrier is placed between the dot layer and the SCH layer at the n-side, carrier injection takes place by tunneling direct to the dot layer. Kamath and his colleagues examined the optical gain and

300

HIROSHIISHIKAWA

the small signal modulation response in both types. In the SCH type, the optical gain measured by the Hakki-Paoli method in the 4-layer stacked quantum-dot laser was 2.5 x 10-'4cmz, which was larger than in the single-layer laser. Nevertheless, there was no significant increase in the modulation bandwidth, which was limited to 6.2 GHz. The tunnel injection laser showed a larger differential gain of 6 x 10-'4cm2, however, the bandwidth was 8.5 GHz. Also, the increase in bandwidth did not correspond to the increase in differential gain. The researchers concluded that the bandwidth was limited by the carrier relaxation, or the phonon bottleneck. This result fits with our discussion that slow carrier relaxation gives a large value of E x to limit the modulation bandwidth. A similar result was reported by Mao et al. (1997). They measured the relaxation oscillation frequency of InGaAs/GaAs self-organized quantumdot lasers, observing a saturation of the relaxation oscillation frequency despite an increase in optical power. The relaxation oscillation frequency saturated at 5.3 GHz at the optical power of 9.1 mW, which also suggests a large value for the gain compression factor, 2, and the existence of the phonon bottleneck. Wang et al. (1997) reported a comparison of wavelength chirping in quantum-well, quantum-wire, and quantum-dot lasers under an ultra-short optical pulse excitation. They fabricated InGaAs/InGaAsP DFB-type lasers emitting at 1.4pm using the etching and regrowth technology. Their quantum-dot diameter was 60 nm, which is a little too-large for sufficient three-dimensional carrier confinement. The researchers reported that the wavelength chirp was smaller in the higher-dimensional carrier confinement, as shown in Fig. 7.3. In quantum-well and quantum-wire lasers, lasing under pulsed excitation starts at a carrier density higher than the threshold level. Then the lasing starts at a shorter wavelength. As a depletion of carriers refractive index increases to give longer wavelength lasing, that is, the red chirp takes place. What is interesting is that Wang et al. reported almost zero-frequency chirp in the quantum-dot laser except for a slight blue chirp at the onset. Their explanation was that the chirping behavior in the quantum-dot laser is determined predominantly by the carriers in the barrier layer because of a very small optical confinement to the dot layer of 0.5%. The carrier relaxation process, which is due mostly to the Auger scattering of carriers, generates high-energy carriers in the barrier layer. This causes carrier heating, and which might cause a chirping different from that in the quantum-well or the quantum-wire laser. Wang et al.'s interpretation did not refer to the c( parameter of the quantum dot, and it considered the effect of carriers in the barrier layer to be the cause of anomalous chirp behavior. This finding is somewhat different from what I have discussed in this section. However, their result is encouraging in that almost zero chirp was obtained.

7 APPLICATIONS OF QUANTUM DOTTO OPTICAL DEVICES

301

Time delay (ps)

FIG. 7.3. Wavelength chirp of quantum-well, quantum-wire, and quantum-dot lasers. Three solid lines represent lasing optical output under optical pulse excitation. The dotted lines represent the change in the lasing wavelength (Wang et al., 1997, 0 1997, Optical Society of America).

Summary Reported experimental results suggest the existence of the phonon bottleneck, which limits modulation bandwidth through the @ term in Eq. (7.15) and through the rolloff given by Eq. (7.16). This is the source of the unresolved problems in high-speed modulation of quantum-dot lasers. To realize the potential of the quantum-dot laser as a high-speed light source, it is necessary first of all to reduce the inhomogeneous broadening to obtain a large differential gain. Also, it is extremely important to clarify the carrier dynamics. We must know the values of x and 2, and we must explore the design of quantum dots to obtain a small value of the product, 2 ~ High . dot density would help reduce the distance between dots to obtain a small diffusion time and a large differential gain. Quantum-dot lasers are worth the efforts to solve these problems because of the potential for zero-chirp high-speed modulation and their potential for very low threshold current operation.

302

HIROSHIISHIKAWA

3. THEQUANTUM-DOT INTENSITY MODULATOR Use of the quantum dots as an intensity modulator was proposed by Sahara et al. (1996) to realize a modulator with a negative a value. In quantum dots grown in the SK growth, we can obtain a Stark shift by applying an electric field parallel to the growth direction. Shown in Fig. 7.4 are the absorption spectra of the quantum dot assumed by Sahara. He assumed a Gaussian-shaped spectrum with a broadening of 34 meV in C 2 full width. This broadening is due to inhomogeneity in the size and composition of the dots. Between two absorption peaks, is a region with a small absorption coefficient due to the discrete energy levels. Adjusting the signal wavelength to the maximum of the absorption peak without application of the field (1.55 pm in Fig. 7.4), we can obtain transmission when the electric field is applied. In this case, as the transmission wavelength is on the shorter wavelength side of the absorption peak, the change in the refractive index gives the negative a value. We can obtain the wavelength chirp to compress the optical pulse in the fiber transmission. Figure 7.5 shows the calculated ci parameter and the change in the absorption coefficient as functions of the applied electric field. In the calculation, the Stark shift caused by the application of the electric field was assumed to be the same as that of the quantum well. The a parameter is negative for a high electric field of 130 to 80 kV/cm, and it shifts to positive at lower fields. In the operation of this modulator, we apply the electric field to the modulator for light-on state and reduce the field for the light-off state. When we turn off the light, electric field is reduced from the high field to the lower field. At this high field region CI is negative and give the blue

Y

1.45

1-50

1.55

1.60

1.65

1.70

Wavelength (pm) FIG. 7.4. Assumed absorption spectra of quantum dots. An inhomogeneous broadening of 34 meV (e-’ full width) is assumed. The dotted line represents the absence of the electric field (Sahara et al., 1996, 5Q1996 I E E E ) .

7 APPLICATIONS OF QUANTUM DOTTO OPTICAL DEVICES

303

c A

s c 0

1280-

960-

O

IlO

1w

t30

th 4b io

0

Fleld (kVlcm) FIG. 7.5. Calculated absorption coefficient and the a parameter, after Sahara et al.. 1996, 8 1996 I E E E .

chirping to compress the light pulse. The positive a parameter at low field region is not at all a problem, because the light is almost absorbed by quantum dots under the low field. It was deduced that for a 100-pm-long device with an optical confinement factor of 0.25, we could obtain transmission changes from 0.57 to 0.017, corresponding to a 15-dB extinction ratio. The analysis by Sahara et al. is a preliminary one. The researchers assumed a Stark shift the same as that of the quantum well, but in actual quantum dots a Coulomb interaction forms the exciton or bi-exciton, which might have some effect on the Stark shift. The assumed inhomogeneous broadening of 34meV is based on the SK-growth mode. The progress of growth technology will largely reduce it, and a narrower broadening will largely reduce the absorption loss between discrete levels to give a much larger modulation extinction. At present, practical modulators that make use of the quantum dot are not a realized application, given the problems to be solved. However, they are worth more detailed investigation because of the need for controlled-chirp high-speed modulation for longer-span transmission systems.

IV. Quantum Dots as a Nonlinear Medium

1. THENEEDFOR LARGENONLINEARITY WITH BANDWIDTH

A

LARGE

It has long been a dream to achieve large optical nonlinearity with large bandwidth using semiconductor structures for optical nonlinear devices. There are many applications waiting for this dream to become a reality:

304

HIROSHIISHIKAWA

high-speed all-optical spatial and time domain switches, wavelength converters for WDM (wavelength division multiplexing) systems, conjugate wave generation for dispersion compensation of optical fiber (Watanabe, 1993), and so on. That there are as yet no practical semiconductor-based optical nonlinear devices is due to small nonlinear susceptibility or, even with large susceptibility, small bandwidth. Here, I will explain this situation, taking four-wave mixing as an example referring to the results of Kawatsuka and his colleagues (1995, 1997). The third-order nonlinearity of the semiconductor gain medium, like semiconductor optical amplifiers, can be used for wavelength conversion by four-wave mixing. Usually, four-wave mixing is done by coupling a pump beam and a signal beam to a semiconductor optical amplifier, as shown in Fig. 7.6(a) (for example, Newkirk and Miller, 1993). A new method, illustrated in Fig. 7.6(b), was demonstrated by Kuwatsuka et al. (1995) in which the lasing power of a A/Cshifted DFB laser itself is used as a pump beam, allowing us to get a wavelength-converted conjugate wave just by coupling a signal beam. Figure 7.7 shows an example of measured wavelength conversion efficiency using this method. The measurement employs a A/Cshifted longcavity DFB laser with an MQW active layer. The conversion efficiency is asymmetric with respect to the pump beam wavelength. The conversion efficiency is higher for the positive detuning; that is, the input signal

-

Semiconductor optical amplifier

%

0,

Single-mode semiconductor laser

(b) FIG.7.6. Four-wave mixing in a semiconductor gain medium: (a) an optical amplifier to which both the pump wave and the signal wave are coupled: (b) a single-mode semiconductor laser, in which the lasing beam itself is used as a pump wave.

7 APPLICATIONSOF QUANTUM DOTTO OPTICAL DEVICES

s =

0

305

' " " " " ; ' " " " " ~

>r

g

aa ..-0

-10

=aa -20 E

.-02

aa >

-30

5 -40

1500

1520 1540 1560 1580 1600 Signal wavelength ( nm )

FIG. 7.7. Example of a conversion efficiency from a signal to the conjugate wave.

wavelength is on the longer wavelength side of the lasing wavelength and is smaller for the negative detuning. Shown in Fig. 7.8 is the third-order nonlinear susceptibility, x’~’, evaluated from the conversion efficiency for the positive detuning. The horizontal axis is the detuning between the pump beam and the signal beam. There are two components in xt3).One is the large x’~’ at low-frequency regions that is due to the carrier density beating whose time constant is a recombination lifetime, and that has rather low response speed. The other is the f 3 ) that is the spectral hole burning effect. It has a very large bandwidth of 6THz, which corresponds to 50fs of dephasing time. This time constant is due to carrier-to-carrier scattering. The asymmetry in the conversion efficiency ) the carrier-densityarises from the difference in the phase of x ' ~ between beating component and the hole burning component. For positive detuning these two components are in phase to give the larger f 3 ) , while for negative detuning these components are out of phase to give the smaller total x(3' (Ogasawara and Ito, 1988). The fast component of f 3 ' is attractive for very-high-speed nonlinear optical device applications; however, the value is not large enough. Generally in nonlinear mediums, the larger the bandwidth, the smaller the nonlinear susceptibility. The wavelength conversion using four-wave mixing with a sufficient signal-to-noise ratio was realized only for a small wavelength detuning of several nms. The fiber dispersion compensation by phase conjugation using the four-wave mixing in the E./it-shifted DFB laser was demonstrated for a small detuning of 2.5nm (Watanabe et al., 1997).

!

306

HIROSHIISHIKAWA

10-14,

r

10'2 1013 1014 Frequency detuning, Aw (radls) FIG.7.8. Third-order nonlinear susceptibility of MQW for positive detuning estimated from the conversion efficiency. The dotted lines represent the experimental result. f 3 ' has two components: the carrier density beating effect and the hole-burning effect.

Asymmetry in the conversion efficiency is another problem for practical application. Using quantum dots to obtain a large x’~)with a sufficiently fast response speed and no asymmetry will be a breakthrough for practical nonlinear optical devices. Such applications will not be limited to four-wave mixing. In Section 11 we saw that we can expect an almost zero refractive index change at the resonant wavelength in quantum dots. However, at the off-resonant wavelength or at the shoulder regions of the inhomogeneously broadened emission line, we can expect a large change in the refractive index by the change in the pump power owing to the large x’~). We can use this refractive index change to make all optical switches. Here, I try to calculate the third order nonlinear susceptibility of quantum dots to see if using them as a nonlinear medium is feasible.

2. ANALYSISOF x ( ~ ) Framework of the Analysis

In Section 11, I discussed the third-order nonlinear susceptibility of quantum dots. The key determinant of the amplitude and the response speed

7 APPLICATIONS OF QUANTUM DOTTO OPTICAL DEVICES

307

of f 3 ) is the dephasing rate, yo,. We want a dephasing rate that gives a sufficiently fast response-say an order of THz-but not so fast as to reduce the amplitude of x(~), as in the case of the spectral-hole-burning component in a quantum well. So far discussions on carrier dynamics in quantum dots have centered on carrier energy relaxation, which is related to the phonon bottleneck problem (Bockelmann and Bastard, 1990; Bockelmann, 1993; Efros et al., 1995; Uskov et al., 1997; Uskov et al., 1998). However, the fast component of is determined not by the energy relaxation but by the dephasing rate, y,,. Figure 7.9 shows several dephasing processes for a carrier in the ground state of a quantum dot. Dephasing takes place due to thermal excitation, recombination, electron excitation through Auger like process, and carrier exchange scattering process between carriers in the ground state and in the barrier layer. Among these four processes, the thermal excitation rate is not very high when the sublevel separation energy does not coincide with the LO-phonon energy. The recombination rate is on the order of lo9- lO'O/s, which is also not very high. The Auger like carrier excitation rate depends on the carrier density at the high energy region and does not seem high, although the inverse of this process is important in considering carrier energy relaxation. Through the Auger like process carriers are supplied to the ground state from the barrier region (Efros et al., 1995; Uskov et al., 1997; Uskov et al., 1998). The highest dephasing rate may be obtained for

0

l---i-r i-fr

Thermal excitation

---*--

Recombination

Auger-like excitation

--om-

Auger-like carrier exchange

FIG. 7.9. Dephasing processes in a quantum dot.

308

HIROSHIISHIKAWA

the carrier exchange process provided when the carrier concentration in the barrier layer is high. The e-e or h-h scattering through the Coulomb interaction causes the exchange of a bound electron/hole and a free electon/hole. To explore the feasibility of third-order nonlinear susceptibility of quantum dots, I perform a trial calculation, taking the dephasing rate due to the carrier exchange as a major cause of dephasing. I introduce some rough approximations and estimate the third-order nonlinear susceptibility using Eq. (7.2) of Section 11.

CalcuIation of the Dephasing Rate due to Carrier Exchange For the sake of simplicity, I make the following assumptions and approximations. 1. Spherical quantum dot is embedded in a bulk large-band-gap material. 2. The Carrier wave function in the barrier bulk material is a plane wave and remains so even at the quantum dot. 3. As an interaction Hamiltonian, we use screened Coulomb potential with a Debye-limit screening parameter (Sinha and DiDomenico, 1970). 4. Dielectric constants are constant, not dependent on the wave number and energy. For the spherical quantum dot, the ground-state wave function is


(7.17)

Itl\ 2 u

(7.18)

Yd(rl)= A sin(ctr,)/r,

for Ir,l

Yd(rl)= ACexp(-Br,)/r,

for

where A and C are normalization constants. We assume that a plane wave is incident on the quantum dot. Thus, the initial-state wave function is the product of the incident plane wave and the dot wave function,

Yi(rl,r2) =

1 ~

exp(ik*r,)Y’,(r ,)

Jn

(7.19)

where, r l and r2 are the respective position coordinates of the incident

7

APPLICATIONS OF QUANTUM DOTTO OPTICAL DEVICES

309

carrier and the ground-state carrier. The final state wave function is YJ(rl, rz) =

1

Jn

~

exp(ik'-rl)Yd(rz)

(7.20)

This means that the incident electron is captured at the ground state of the quantum dot and the bound electron in the dot is excited, becoming a free electron with a wave number of k'. The interaction Hamiltonian for the process is a screened Coulomb potential, as

where e is the electron charge, E is the dielectric constant, and 9 is a screening parameter, which is given as a function of carrier density in the barrier layer as 9=

(qz

(7.22)

where k is the Boltzmann constant and T is the absolute temperature. The matrix element, M , for the process is

This matrix element can be evaluated by the Fourier transform, Fd(k), of the ground-state wave function as

(7.24) Then

-

d3k"

e-~k'.r,~;(kt!)e-Ik".r,Herk.r, Fd(k"')e'k"'~'~d3r (7.25)

310

HIROSHIISHIKAWA

Assuming that the band is isotropic in K space, the scattering rate is given by Fermi’s golden rule as w =2 . 2

h

jk

d3k

I, ($r \MI2

f(k)(l - f(k))G(E - Ef)d3k’ (7.26)

where the initial-state energy, E i , and the final-state energy, E,, are given by (7.27) and f(k) is the Fermi-Dirac distribution function. Factor 2 in the scattering rate is due to spin.

Numerical Calculation and Results The numerical calculation assumed an Ino,,3Gao,,7As spherical dot embedded in bulk InP. Numerical parameters are as follows: Potential depth for the electron: 270 meV Effective mass of the electron: 0.044m0 Potential depth for the hole: 360 meV Effective mass of the hole 0.54m0 Diameter of the spherical quantum dot: 20 nm Dielectric constant: 3.22 x to Potential depths were calculated following the method of Sugawara (1993). We assumed for simplicity that the carrier masses were equal within the well and the barrier. Figure 7.10 shows the scattering rate at 300 K. Scattering due to e-e and h-h and the total rate are shown as functions of carrier density in the bulk barrier of InP. The scattering rate of the hole is larger at a higher carrier density because of the larger density of states in the barrier layer for holes. As can be seen from Fig. 7.10, we can obtain a total scattering rate on the order of l O I 3 for the carrier density of the barrier layer larger than 5 1017/~~3. The third-order susceptibility was calculated using Eq. (7.3) in Section 11, where ycc is the e-e scattering rate, yI., is the h-h scattering rate, and y,, is its total. We assumed the inhomogeneous broadening of 6 meV and 30 meV. Inhomogeneous broadening was incorporated by calculating the groundstate transition energies for the Gaussian distribution in the diameter of

7 APPLICATIONSOF QUANTUM DOTTO OPTICAL DEVICES 1oi4

1

I

I

311

I

300 K

109'

lo**

I 1015

I

1016

I

10'7

I

10'8

I

Carrier density in barrier ( ~ r n - ~ ) FIG. 7.10. Calculated exchange scattering rate as a function of carrier density in the barrier layer.

quantum dots, and by summing for all dots. The dot density was at 2 x 1016/cm3,which is equivalent to the volume of 8%, The result is shown in Fig. 7.11 as a function of the angular detuning frequency between the pump and the signal wave. Results are shown for carrier densities in the barrier layer of 1.85 x 10"/cm3 and 1.94 x 10"/cm3. The ycc, yo,, and y,, for the carrier concentration of 1.85 x 1017/cm3 are 1.1 x lo", 4.08 x lo", and 5.18 x 101's-', respectively, and, for 1.94 x 101'/cm3, 5.03 x lo", 1.107 x and 1.159 x 10'3sp1,respectively.

3. DISCLJSS~ON A rough estimate of the third-order nonlinear susceptibility was made for a simplified model, taking the carrier exchange scattering process as a cause of dephasing. Our model is, of course, incomplete, but we can still obtain perspectives for the third-order nonlinearity in quantum dots.

Dephasing Rate There is a possibility of realizing a dephasing rate of 1 x 1013s - l by use of exchange scattering. Of course, other scattering mechanisms may further

312

HIROSHI ISHIKAWA

10-13

I

FWHM=6meV

- 1 . k x 1017 crn-a

--.1.94 x 1018m-a

-N

>

N -

E

Y I

L

X

Detuning, Aw (mas) FIG.7.11. Calculated wave.

x ‘ ~ as ’ a function of detuning between the pump wave and the signal

shorten dephasing time, however, from the measurement and analysis of the is estimated to be 10-loops (Mukai, real carrier energy relaxation, tlelar 1995; Mukai et al., 1996b; Uskov et al., 1997; Uskov et al., 1998). The modulation bandwidth is limited to 5-8 GHz (Kamath et al., 1997; Mao et al., 1997; Wang et al., 1997). These estimates indicate that dephasings associated with the energy relaxation is not so fast. Exchange scattering could be an important way to quicken the response of quantum dots when the carrier density in the barrier layer is high. Also interesting is that our dephasing rate depends on the quantum dot’s geometry. This can be seen from the functional form of the matrix element, which is

where k”’ is the wave number of the ground-state carrier, and k is the wave number of an incident carrier. This equation means that the maximum is obtained when the K-space distribution of the ground-state electron wave function reaches the wave number of the incident electron. Because the K-space distribution of IFd(k”’)ldepends on the geometry of the quantum

7 APPLICATIONSOF QUANTUM DOTTO OPTICAL DEVICES

313

dot, the dephasing rate may depend on the geometry. Our calculation is for a spherical dot, so it will be interesting to extend the calculation to SK dots whose geometry is not spherical.

Amplitude and Bandwidth of

x(3)

When compared with the f 3 ' for 1.85 x 1017/cm3 and 1.94 x 1018/cm3, the bandwidth is narrower and the amplitude is smaller for higher concentration of 1.94 x lOl8/cm3 as can be seen from Fig. 7.11. Narrower bandwidth for higher concentration is because of the smaller dephasing rate of electrons at very high concentration regions. The smaller amplitude of x(3) is because of the larger total dephasing rate. At a lower carrier density of 1.85 x lOl7/crn3, both electrons and holes have the appropriate dephasing rate to produce a wider bandwidth close to 1 THz. When compared with the hole-burning component of x’~’ of the quantum well in Fig. 7.8, the bandwidth is smaller because of the smaller dephasing rate. However, a much larger x(3) was obtained even for the inhomogeneous broadening of 30meV. For the broadening of 6meV, x’~)is still one order of magnitude larger than that of the 30-meV broadening. There are two reasons for this. One is the discrete energy levels of the quantum dot, which concentrates oscillator strength at the resonant wavelength. The other is the comparatively longer dephasing time than that of the MQW. The longer the dephasing time, the larger the xt3). What is important is to obtain an appropriate dephasing time to give a sufficiently fast response for device . calculation shows application without decreasing the amplitude of x ' ~ )Our that it is possible to achieve such a condition. Of course, as x’~) is proportional to dot density, an increase of dot density is essential.

Asymmetry in

f3)

The asymmetry in x’~’with respect to the pump wavelength is a problem in the f 3 ' of carrier-injected MQW. It arises from the phase difference between the carrier-density-beating effect and the hole-burning effect. In quantum dots, we can expect a very small change in the refractive index for the change in carrier concentration near the inhomogeneously broadened resonant wavelength, as discussed in Section 11. We can expect an in-phase response of the carrier-density-beating effect and the hole-burning effect near the resonance. Although there may be some effect from the carriers in the barrier layer, we should see an almost symmetric x’~’with respect to the pump wavelength.

314

HIROSHIISHIKAWA

Thus, with a simplified model and analysis, I have calculated the x(3) of quantum dots, and I have demonstrated the possibility of obtaining fast value than that of responses of around 1 THz, and also a much larger quantum wells. In addition, I have shown the possibility of obtaining a symmetric f 3 ) with respect to pump wavelength. Of course, development of more precise and rigid theory is required to draw decisive conclusions, especially for the dephasing process. Our discussion is for a four-wave mixing process, but prospective applications are not limited to that. I emphasized the small refractive index change close to the center of the inhomogeneously broadened spectral line, which should eliminate asymmetry in f 3 ’ . However, at the shoulder regions of the inhomogeneously broadened line, we can expect a large change in the refractive index due to the large x(~).Possibly we can control the refractive index at the shoulder region by the pump power. There will be applications for all optical switches. So far, no concrete measurement of third-order susceptibility has been reported. Experimental verification of the usefulness of the quantum-dot’s nonlinearity is a very important task, together with development of a more rigid and precise theory of nonlinearity.

V. Persistent Hole Burning Memory

The inhomogeneous broadening of quantum dots is an obstacle to obtaining large linear gain or large non-linear susceptibility. However, Muto (1995) proposed a use of it as a persistent hole burning (PHB) memory. Here I review Muto’s idea and then introduce recent experimental results.

1. PERSISTENT SPECTRAL HOLEBURNING MEMORY USING DOTS QUANTUM

Persistent spectral hole burning memory is highly attractive because it makes feasible the storing of information with a very high multiplication factor in a small area by use of the wavelength domain. At a low temperature of 4 K , high multiplication factors (MPFs) of lo3-lo4 are expected, using polymers, alloys and glasses. The problem with such materials is that the M P F reduces at higher temperatures, and it has been very difficult to make practical PHB memories that operate at room temperature (Ao et al., 1993). Muto noticed that inhomogeneous broadening of the ground-state transition can be used for PHB if we can realize

7 APPLICATIONS OF

QUANTUM

DOTTO OPTICAL DEVICES

315

persistent hole burning in quantum dots. The inhomogeneous broadening so far reported is around 25-80 meV (for example, Ishikawa et al., 1998) and can be increased to 100meV by adjusting the growth condition. If the homogeneous linewidth of the ground-state transition is below 0.1 meV, the M P F can be on the order of lo3. Muto proposed a method of obtaining persistent hole burning by extracting electrons from the potential well of the conduction band. When a quantum dot is irradiated by a ground-state electron-hole transition wavelength, the electron is excited to the ground state of the conduction band. If it is left as is, recombination takes place and no persistent hole burning occurs, but if we can extract electrons from the ground state-for example, by tunneling to the X band of the barrier material-the hole remains in the ground state and there is no additional absorption. In other words, persistent hole burning takes place. Muto gives us a more detailed estimation of the performance of the PHB memory using quantum dots. The memory duration is determined by the product of the probability of electrons going back to the ground state and the recombination rate. Muto assumed the probability to be the FermiDirac distribution and further assumed 20 ns of recombination lifetime. Thus, he obtained a duration time of 0.2ms at room temperature. To get much longer memory time, some refreshing procedure would be necessary. Muto also made some estimation for the reading. He assumed the reading by photo-induced current by applying the reading photon flux. The condition for reading is a sufficient signal-to-noise ratio (S/N) without the reading photon flux burning the “virgin”, or unburned, wavelength. The signal in reading is given by the difference between the currents transmitted through the virgin area and those transmitted through the marked area. The noise is defined as a sum of the root mean square of fluctuation of shot noise due to the currents at the virgin region and the burned region. Muto required an S/N larger than 20, giving the minimum of the reading photon flux, F,. Another condition that photon flux does not burn the virgin region is that T ~smaller than unity, where D is the absorption the product of D F ~ be cross-section and z R is the reading time. This gives the maximum of the cross-section, D. The maximum MPF is defined to be N,,,/N,, where N,,, is the total density of the dot and N, is the density of dots within a homogeneous linewidth at o.Note that the total absorption coefficient, is DN,, (7.29) where L is the absorption length. Assuming a realistic homogeneous broadening of 1 meV, an D of 1 x 10- cm2, and a n area of reading laser beam

’

316

HIROSHIISHIKAWA

of 1pm’, Muto obtained MPFIL < 6.7 x 10/pm

(7.30)

to satisfy the condition aFRzR< 1. For an N,,, of 1 x 1018/cm3,an homogeneous broadening of 1meV and an inhomogeneous broadening of 200meV, M P F is 100 for a 1.5-pm-thick absorption layer. In this case, 1.5 x lo4 quantum dots contribute one bit of memory.

2. EXPERIMENTAL RESULTS Several experiments confirm the persistent hole burning effect in quantum dots. One used a Shottkey contact structure, shown in Fig. 7.12 (Imamura et al., 1995). The i-GaAs region contained self-organized InAs quantum dots with an inhomogeneous broadening of 73 meV. When the structure was irradiated by the writing optical pulse, an electron-hole pair was generated at quantum dots whose transition wavelength coincided with the write pulse. Because of the built-in field caused by the Shottkey contact, electrons excited out to the conduction band tunnels from the dot, while the holes remained at the ground state because of the high barrier. This bleached the quantum dots. Imamura and his colleagues measured the difference of the photo-induced current for the write and read pulses, which was At after the write pulse at a room temperature of 300 K. They used the same wavelength for both write and read. The result is shown in Fig. 7.13. For the small time difference, the difference in the current was large, which means that persistent hole burning was taking place. The difference reduced for a larger At, which Imamura

FIG. 7.12. Band lineup of a sample used in the persistent hole-burning experiment by Imamura et al. (1995, 0 1995 J p n . J . Appl. Phys.).

7

APPLICATIONS OF QUANTUM DOTTO OPTICAL DEVICES

lo-’

I

I

I

I

I

I

l

lnAs QDs, 300 K

317

i

-

Q) c .

L

-

I I I t I I I 10-0 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

At (ms) FIG 7 13 Difference in the photo-induced current between the write pulse and the read pulse as a function of a time interval between the two pulses (Imamura et al , 1995, P 1995 J p n J Appl P h y s )

attributed to the recombination of residual holes with thermally excited electrons from the n-GaAs. From the experimental result, Imamura found the retention time of the hole burning to be 0.27 ms at room temperature. Sugiyama and his colleagues (1997) observed the spectral hole at a low temperature of 5 K. Their sample, shown in Fig. 7.14, had a p-i-n structure

I

n+-GaAs,300 nm (001) n+-GaAssubstrate

1

I

FIG. 7.14. Schematic of a sample used in the persistent hole-burning experiment by Sugiyama (1997, Q 1997 I E E E ) .

318

HlROSHl ISHlKAWA

1 x lo-”

9 U

e

0

2

-1 x lo-” 1063

1084

106S

1066

Wavelength (nm) FIG. 7.15. Comparison of experimental and calculated results of spectral hole burning at 5 K: (a) the experimental result, (b) a calculated fit. The writing wavelength of YAG consists of three wavelengths, as indicated by the arrows (Sugiyama et al., 1997, 0 1997 IEEE).

with five-layer stacked quantum dots at the i region. Each layer contained dots with a density of 5 x 10’o/cm2. The sample was irradiated by a CW YAG laser at 1064 nm, and the reading was taken using a tunable CW TiS laser. The YAG laser consists of several emission lines 0.1 or 0.2 nm apart with a linewidth of 27 peV. The linewidth of the TiS laser is below 165 peV. Measurements were performed at 5 K. After irradiation by the YAG laser at 8-mW output, the photoresponse current was measured irradiating by the TiS laser at 30pW. When the reverse-bias voltage to the sample was below 4 V, the researchers observed no spectral hole in the response current. However, when they increased the inverse bias up to 5 V , an apparent spectral hole became visible. Trace (a) in Fig. 7.15 is the measured spectral hole under 6-V reverse bias. At a low bias voltage, the generated electron and hole remained in the dots and recombined, producing no spectral hole. When the bias voltage was increased, the electron tunneled out from the dot, initiating persistent hole burning. This took place for a bias voltages larger than 4 V. Sugiyama and his colleagues simulated the shapes of the spectral hole. Assuming a Gaussian-shaped TiS read light with an FWHM of 165peV, and assuming a Lorentzian-shaped spectrum for both the YAG read light and the quantum dot, they performed the fit with a measured spectral hole, taking convolutions. Trace (b) in Fig. 7.15 shows the calculated result. Note

7

APPLICATIONS OF QUANTUM DOTTO OPTICAL DEVICES

319

that the YAG emission line consists of three lines, as indicated by arrows in the figure. The best fit was obtained for a homogeneous broadening smaller than 80 peV.

3. DISCUSSION Based on Muto’s proposal, experiments were performed to verify the hole-burning effect in quantum dots. The existence of spectral holes was confirmed at 300 K, with a retention time of 0.27 ms, and the spectral-hole shape was measured at a low temperature of 5 K. These results encourage us to believe that persistent hole burning memory using quantum dots is feasible. The maximum multiplication factor depends largely on the parameters of quantum dots, such as homogeneous broadening width, dot density, and dot-layer thickness. If we can achieve an M P F of 100 for a 1-pm2 area, we will have 10Tb/cmz of memory. This is a huge capacity. For this reason, persistent hole burning memory is a highly attractive application of quantum dots.

VI. Summary and Perspectives on Quantum-Dot Optical Devices

In this chapter, I reviewed the basic properties of quantum dots and discussed several possible device applications. I also discussed chirpless high-speed modulation of quantum-dot lasers, a blue-chirp intensity modulator, the feasibility of quantum dots as a nonlinear medium, and state-ofthe-art research of Tb/cm2 persistent hole burning memory. Of course, there are many other applications for quantum dots. Among them is an infrared photodetector which, although not discussed here, is considered an attractive alternative to the quantum-well infrared photodetector (QWIP) (Ryzhii, 1996). Our discussion in this chapter was not necessarily based on concrete theory. My treatment of the quantum dot as a two-level system is insufficient under high optical power, and my analysis of the dephasing rate is merely a rough sketch of the process that may require the inclusion of phonon scattering to be more comprehensive. Many more detailed and concrete theoretical studies, together with experimental verification, are needed. Nevertheless, we have obtained some perspective on quantum dots, both positive and negative, for device application.

320

HIROSHIISHIKAWA

Finally, I sketch the trends of optoelectronics and illustrate some uses for quantum-dot optical devices.

1. TRENDSIN OPTOELECTRONICS Research on optoelectronics has been fueled by a strong need for optical communication systems. At present, long-haul, local-area, and undersea cable systems are in practical use. Moreover, applications are expanding to systems not only for communication but for information processing as well. Still larger capacity and/or longer-distance trunk line and undersea cable systems are being developed. Local-area and access network systems are also urgently required. The optical interconnection between boards and chips, and within chips, is becoming a research target. Other than communication and data transmission, optical data storage and optical image processing are also important research areas. Two approaches are being taken to achieve larger-capacity transmission. One is the WDM system; the other is the much higher data rate TDM (time division multiplexing) system. WDM systems with a multiplication factor of 30-80 wavelengths (with spacing of 0.4-0.8 nm), each carrying a 2.5-10Gb/s signal, are being developed. They are attractive not only for their large multiplication but also for their flexible routing of signals by wavelength. However, these systems require the development of new devices such as fixed and/or tunable wavelength filters, multi-wavelength and/or tunable light sources, and wavelength conversion devices and those with a very high bit rate of above 10 Gb/s demand very low chirp, high-speed light sources. 40 Gb/s systems are being planned, and Tb/s TDM systems are envisioned. In both WDM and TDM systems, dispersion compensation of a fiber to enable long-distance transmission is essential. Access networks systems, such as fiber to the home (FTTH) and fiber to the curb (FTTC), require low power consuming lasers that provide temperature-robust performance. For optical interconnections, we must develop very low power consuming, temperature-robust light sources for parallel optical links between computers, boards, chips, and, hopefully within chips. The problems of wiring and pin bottlenecks should be solved by the introduction of optics in interconnections. However, very cheap, high-performance devices are required in access networks and interconnections, especially optical interconnections. These devices must be cheaper than copper-wire interconnections and must have sufficiently higher capacity data throughputs. In all of the above systems, space, time, and wavelength domain optical switches will be essential to achieve flexibility and smartness. Data storage also will be important for small-size and huge-capacity throughput.

7 APPLICATIONS OF QUANTUM DOTTO OPTICAL DEVICES

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2. USESFOR QUANTUM DOTOPTICAL DEVICES

Quantum dots will play an important role in the future of optoelectronics. For access networks and interconnections, the quantum-dot laser could be a key component because of its potential for low threshold current operation and temperature-robust operation (see Chapter 6), but keeping them low-cost will take extensive effort. For high-speed TDM systems above 10 Gb/s, the very small wavelength chirp under high-speed modulation in quantum-dots is attractive, as is the blue-chirp quantum-dot intensity modulator. For WDM systems, the nonlinearity of quantum dots looks promising as a wavelength conversion device. Generation of a phase conjugate wave by a nonlinear process such as four-wave mixing can be used for dispersion compensation of optical fibers to attain high bit rate long-distance transmission. The quantum dot’s nonlinearity may also be used in space and time domain switches to increase their flexibility and smartness. Finally, quantum dots have the capacity for large data storage because of the persistent hole burning effect, and the potential for low-dark current infrared photodetection and imaging. Thus, we see many important roles for quantum dots in optoelectronics. As I said before, however, we must establish low-cost, high-quality fabrication technologies for their practical application. It is my hope that some of the devices discussed here will come into practical use within the next five years.

Acknowledgment The author would like to give hearty thanks to his collaborator, Dr. H. Kuwatsuka, for his comments and advice. The study of Section IV quantum dots as a nonlinear medium has been done under the management of Femtosecond Technology Research Association (FESTA), supported by the New Energy and Industrial Technology Development Organization (NEDO) Japan.

REFERENCES Adachi, S. (1982). 1. Appl. Phys. 53, 5863. Ao, R., Kummerl, L., Haarer, D. (1993). J p n . J . Appl. Phys. 32, 5248. Bennett, B. R., Soref, R. A., Del Alamo, J . A. (1990). IEEE J . Quantum Electron. 26, 113 Bockelmann, U., and Bastard, G . (1990). Phys. Rev. B. 42, 8947. Bockelmann, U. (1993). Phys. Rev. B. 48, 17637.

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Bowers, J. E., Hemenway, B. R., Gunack. A. H., and Wilt, D. P. (1986). IEEE J . Quantum Electron. QE-22, 833. Devaux, F., Bigan, E., Ougazzaden, A., Huet, F., Carre, M., and Carenco, A. (1992). Electron Lett. 28, 2157. Efros, A. L. Kharchenko, V. A., and Rosen, M. (1995). Solid State Comm. 93, 281. Green, C . A., Dutta, N. K., and Watson, W. (1987). Appl. Phys. Lett. 50, 1409. Haisch, H., Baumert, W., Hache, C., Kiihn, E., Klenk, M., Satzke, K., Schilling, M., Weber, J., and Zielinski, E. (1994). Proc. 201h Eur. Con$ Optical Communication 2. Henry, C. H. (1982). IEEE J . Quantum Electron. QE-18, 259. Hu, Y. Z., Lindberg, M., and Koch, S. W. (1990. Phys. Rev. B. 42 1713. Imamura, K., Sugiyama, Y., Nakata, Y., Muto. S., and Yokoyama, N. (1995). Jpn. J . Appl. Phys. 34, L1445. Ishikawa, H., Soda, H., Wakao, K., Kihara, K., Kamite, K., Kotaki, Y., Matsuda, M., Sudo, H., Yamakoshi, S., Isozumi, S., and Imai, H. (1987). IEEE J. Lightwave Technol. LT-5,848. Ishikawa, H., and Suemune, I. (1994). IEEE Photonic Technol. Lett. 6, 1315. Ishikawa, H., Shoji, H., Nakata, Y., Mukai, K., Sugawara, M., Egawa, M., Otsuka, N., Sugiyama, Y., Futatsugi, T., and Yokoyama, N. (1998). J . Vacuum Sci. & Technol. A16,794. Kakimoto, S., Nakajima, Y., Sakakibara, Y., Watanabe, H., Takemoto, A,, and Yoshida, N. (1990). IEEE J . Quantum Electron. 26, 1460. Kamath, K., Klotzkin, D., and Bhattacharya. P. (1997). IEEE Lasers and Electo-Optics SOC. Ann. Meet. ThY4, San Francisco. Kane, E. 0. (1957). J . Phys. Chem. Solids 1. 249. Kataoka, T., Miyamoto, Y., Hagimoto, K.. Sato, K., Kotaka, I., and Wakita, K. (1994). Electron Lett. 30, 872. Kirstaedter, N., Ledentsov, N. N., Grundmann, M., Bimberg, D., Ustinov, V. M., Ruvimov, S. S., Maximov, M. V., Kop’ev, P. S., Alferov, Z. I., Richter, U., Werner, P., Gosele, U., and Heydenreich, J. (1994). Electron. Lerr. 30. 1416. Koch, T. L., and Linke, R. A. (1986). A p p l . Pkys. Lett. 48. 613. Koyama, F., and Iga, K. (1985). Elerrron. Lerr. 21, 1065. Kuwatsuka, H., Shoji, H., Matsuda, M.. and Ishikawa, H. (1995). Electron Lerr. 31, 2108. Kuwatsuka, H., Shoji, H., Matsuda, M., and Ishikawa, H. (1997). I E E E J . Quantum E/ectron. 33, 2002. Lee, Y. H., Chavez-Pirson, A,, Koch, S. W.. Gibbs, H. M., Park, S. H., Morhange, J., Jeffery, A,, and Peyghambaridn, N. (1986). Phys. Rev. Lett. 57,2446. Mao, M.-H., Heinrichsdorff, F., Kitstaedter, N., Krost, A,, Ledentsov, N. N., and Bimberg, D. (1997). IEEE Lasers and Electro-Optics Sor. Ann. Meet. ThY5, San Francisco. Morito, K., Sahara, R., Sato, K., Kotaki, Y . (1996). IEEE Phofonic Technol Lett. 8. 431. Mukai, K., Otsuka, N., Shoji, H., and Sugawara, M. (1995). Phys. Rev. B. 54, 5243. Mukai, K., Ohtsuka, N., and Sugawara, M. (1996a). J p n . J . Appl. Phys. 35, L262. Mukai, K., Ohtsuka, N., Shoji, H., and Sugawara, M. (1996b). Appl. Phys. Lett. 68, 3013. Muto, S . (1995). Jpn. J . Appl. Phys. 34, L210. Nagarajian, R., Ishikawa, M., Fukushima. T.. Geel, R. S., and Bowers, J. E. (1992). I E E E J. Quantum Electron. 28, 1990. Nakata, Y., Sugiyama, Y., Futatsugi. T.. and Yokoyama, N. (1997). J . Cryst. Growth 175/176, 713. Newkirk, M. A,, and Miller, B. I. (1993). Appl. Phys. Lett. 63, 1179. Olshansky, R., Hill, P., Lanzisera, V., and Powazinik, W. (1987). I E E E J . Quantum Electron. 23, 1410. Ogasawara, N., and Ito, R. (1988). Jpn. J . Appl. Phys. Lett. 27, 615. Ryzhii, V. (1996). Semirond. Sci. Technol. 11. 759.

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323

Sahara, R., Matsuda, M., Shoji, H., Morito, K., and Soda, H. (1996). IEEE Photonic Tecliriol. Lett. 8, 1477. Seki, S., and Yokoyama, K. (1994). Phys. Ren B. 50, 1663. Seino, M. (1996). 1st Optoelectronics and Cornmunicutions Conj.' (Technical Digest) 17D-2, 182. Shoji, H., Nakata, Y., Mukai, K., Sugiyama, Y., Sugawara, M., Yokoyama. N.. and Ishikawa, H. (1996). Electron. Lett. 32, 2023. Sinha, K. P., DiDomenico Jr., M. (1970). Phys. Rev. B. 1, 2623. Soda, H., Sato, K., Nakai, K., Ishikawa, H., Imai, H. (1988). Electron Lett. 24, 1194. Sugiyama, Y., Nakata, Y., Muto, S., Horiguchi, N., Futatsugi, T., Awano, Y., Yokoyama, N. (1997). Electron Lett. 33, 1655. Tucker, R. S. (1985). IEEE J. Lightn~nvr~ Technol. LT-3, 1180. Uomi, K., Tsuchiya, T., Nakano, H., Aoki, M., Suzuki, M., Chinone, N. (1991). I E E E J . Quantum Electron. 27, 1705. Uskov, A. V., Adler, F., Schweizer, H., and Pilkuhn, M. H. (1997). J . Appl. Phys. 81. 7895. Uskov, A. V., Mcinerney, J., Adler, F., Schweizer, H., and Pilkuhn, M. H. (1998). Appl. Phys. Lett. 72, 58. Wang, J., Griesingter, U. A., and Schweizer, H. (1997). Con$ Lusers nd Electro-Optics CWF19, Baltimore. Watanabe, S., Naito, T., Chikama, T. (1993). IEEE Pliotonic Technol. Lett. 5. 92. Watanabe, S., Kuwatsuka, H., Takeda. S., and Ishikawa, H. (1997). Electron. Leti. 33. 316. Wiesenfeld, J. M., Tucker, R. S., and Downey, P. M. (1987). Appl. Phys. Lett. 51, 1307.

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SEMICONDUCTORS AND SEMIMETALS, VOL. 60

CHAPTER 8

The Latest News Mitsuru Sugawara, Kohki Mukai OPTICAL SEMICONDUCTOR DEVICES LAHORATORl FUJITSULABORATORIES Lm. ATSLIGI, KANAGAWA, JAPAN

Hiroshi Ishikawa, Koji Otsubo ELECTRON DOVICOS AND MATERIALS LARoRATORl FLIJITSLI LABORAToRltS LTU. ATSUGI. KANAGAWA. JAPAN

Yoshiaki Nakata QUANTUM ELOCTRON DEVICES LABORATORY

LTD F U J I T SLABORATORIES ~I ATSUUI,KANAGAWA, JAPAN

I. LASING WITH LOW-THRESHOLD CURRENT AND HIGH-OUTPUT POWER FROM COLUMNAR-SHAPED QUANTUM DOTS. . . . . . . . . . . . . . . . 11. EFFECT OF HOMOGENEOUS BROADENING OF SINGLE-DOT OPTICAL GAIN ON LASING SPECTRA . . . . . . . . . . . . . . . . . . . . . . . . . 111. QUANTUM DOTSON INGAASSUBSTRATES . . . . . . . . . . . . . . . . IV. QUANTUM DOTSEMITTING AT 1.3Aim GROWN BY Low GROWTH RATES AND WITH A N INGAASCAP . . . . . . . . . . . . . . . . . . . . . . v. REDUCED-TEMPERATURE-INDUCED VARIATION OF SPONTANENOUS EMWON I N ALTERNATESUPPLY (ALS) QUANTUM DOTSCOVERED BY In,,,Ga,,,As . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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I. Lasing with Low-Threshold Current and High-Output Power from Columnar-Shaped Quantum Dots In the columnar-shaped self-assembled InGaAs quantum dot lasers described in Chapters 2 and 6, we achieved a threshold current of 5.4mA, a 325 Copyright

(

1999 by Academic Press

All rightr of reproduction in any form reserved

ISBN 0-12-752169-0 ISSN 0080-8784/99 $3000

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current density of 160A/cm2, and an output power of llOmW at room temperature (Mukai et al., 1998). Columnar-shaped self-assembled quantum dots are grown by stacking InAs self-assembled dots with a few-monolayer-thick GaAs intermediate layer. The structure as a whole can be considered one physically coupled large dot. This new dot type has high uniformity in size and high emission efficiency, which are desirable properties for semiconductor lasers. Using one columnar-dot layer (n = 1) or two sheets of the layers separated by a 30-nm GaAs layer ( n = 2), we fabricated double heterostructure lasers. The columnar-dot layers were sandwiched between GaAs separate-confinementstructure (SCH) layers. The active region was grown on a 1.4-pm nAl,.,Ga,,,As cladding layer, followed by a 1.4-pm p-Al,.,Ga,,,As cladding layer and a 0.4-pm p-GaAs contact layer. We formed 3-pm-wide and 1.2-pm-high ridge structures by chemical etching.The cavity lengths ( L )were 300 and 900pm. For a laser with n = 2, L = 300pm,and a high-reflective (HR) coating on both facets, lasing oscillation occurred for a threshold current of 5.4mA at 25°C (Fig. 8.1). With L = 900pm and an HR coating on both facets, the threshold current density was reduced to as low as 160 A/cm2 (i.e., 80 A/cm2

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(3 1998 IEEE).

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per single sheet of dot layer). In both lasers, lasing occurred from the top of spontaneous emission of the ground state at 1.15pm. For a laser with L = 900pm and an HR coating on one side of the facets, an output power of 110mW was achieved (Fig. 8.2). Even at 70°C, an output power of 75 mW was obtained. To our knowledge, such low threshold current and high output power have never before been reported for edge-emitting quantumdot lasers. Figure 8.3 shows the lasing emission spectra of the laser with n = 1, L = 900pm, and an HR coating on both facets up to 300mA. The lasing started at the ground state with a threshold current of 11.5mA. As the current increased to 60mA, the lasing spectra became broad with spikes; then, at 80mA, lasing from the excited state appeared at around 1.1 pm. What is surprising is that the lasing at the ground state weakened at 130mA and completely disappeared at 200mA. The spectrum at 300mA shows broad-band lasing only from the excited state. This jump of lasing wavelength from the ground to the excited state can be explained by gain saturation at the ground state, which is due to the phonon bottleneck and/or third-order nonlinearity, and a subsequent increase in the carrier number in the excited state (see Chapter I). The quenching of lasing at the

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0.9

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=

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ground state might be due to the elevated temperature of quantum dots under high current injection and the resultant increase in the carrier emission rate from quantum dots to SCH layers. Let us discuss the reasons that we achieved high lasing performance in columnar-dot lasers. First, high uniformity in size, large density, and high emission efficiency of the columnar dots are probably the decisive factors. Second, closely stacked wetting layers around the dots capture carriers more efficiently than a single wetting layer does, because the wetting layers in columnar dots are electrically coupled like a superlattice and thus have a deep potential (proved by the fact that the emission wavelength is longer than that of SK dots, as shown in Fig. 2.22. Third, the closely stacked wetting layers increase the refractive index around the dots and enhance the optical confinement factor. Fourth, columnar dots with a rather symmetric shape have a a larger electron-hole overlap integral than do ordinary SK dots with a lens-like shape and thus have a higher optical gain.

11. Effect of Homogeneous Broadening of Single-Dot Optical Gain on Lasing Spectra

We found that dots with different energies start lasing independently at low temperatures due to their spatial localization, while at room tempera-

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ture the dots contribute to one-line lasing coilectively via homogeneous broadening of optical gain. The homogeneous broadening at room temperature is estimated to be more than 10meV and can be attributed to the scattering process to exchange electrons (holes) between the ground state and the excited states, including the wetting layers. In Chapter 1, Section V.B, we derived a formula of quantum-dot optical gain that takes into account both inhomogeneous and homogeneous broadening. We pointed out that, if homogeneous broadening is a delta function, each single quantum dot contributes to lasing independently, leading to broad lasing spectra. Also, when homogeneous broadening is comparable to inhomogeneous broadening, the dot ensemble in the cavity contributes to lasing collectively, leading to lasing from the top of the optical gain spectrum, This is confirmed in the experiments on light emission spectra of the columnar-shaped, self-assembled InGaAs quantum dot lasers with a 300-pm cavity and with a high refectivity (95%) coating on both facets. Figure 8.4 compares the light emission spectra between 80 K and 298 K, up to well above the lasing threshold. At 80K, as the current increases, the spontaneous emission rises from the top of the spontaneous emission spectrum, leading to broad lasing emission over the range of 5OmeV at 5 mA and 60meV at 20mA This can be explained as follows: At this low temperature, homogeneous broadening is negligible, and thus dots with different energies have no correlation to each other because they are spatially isolated from each other. As a result, all dots with an optical gain above the lasing threshold start lasing emission independently. The quantum-dot laser in this situation behaves as if it included independent lasing media in the same cavity. At 298 K, the spontaneous emission at 3 mA -about half the threshold current -showed the emission due to the ground-state, band-to-band transition between the discrete quantized states at 1.15pm, the second-state transition at 1.08pm, and the transition at the wetting layers at 1.0pm. At lOmA, the lasing emission line appeared from around the top of the spontaneous emission spectrum. The line at 10 mA has the full width at half maximum of 1.5 nm, and the mode separation of this cavity laser is 0.62 nm, indicating that the line includes 2 or 3 longitudinal modes. At higher injection currents, an additional line appeared at 20 mA with a wavelength of 1.16pm separated by 19meV from the central line at l.l4pm, and then at 40mA with 1.123pm separated by 16meV from the central line. This can be explained as follows: Homogeneous broadening of the optical gain is comparable to inhomogeneous broadening at 298 K, and thus lasing-mode photons come not only from energetically resonant dots but also from other, nonresonant dots that lie within the amount of homogeneous broadening.

MITSURUSUGAWARA ET AL.

330

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FIG. 8.4. Light emission spectra of columnar-shaped self-assembled InGaAs quantum-dot lasers with I I = 2, L = 900pm. and an HR coating on both facets at (a) 8 0 K and (b) 298K.

Since carriers of nonresonant dots are extracted and put into the central lasing modes by the stimulated emission, lasing emission with one line occurs. As the injection current increases, the gain saturation occurs due to the phonon bottleneck and/or third-order optical nonlinearity, which results in new lasing lines in addition to the central line. The energy separation between the lasing lines in Fig. 8.4b- 19 and 16meV-corresponds to the homogeneous broadening of this laser chip. The origin of the homogeneous broadening can be attributed to the

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scattering process to exchange electrons (holes) between the ground state and the excited state, including the wetting layers. According to the theoretical calculation of the scattering rate in Chapter 7, the total scattering rate, this is, the sum of the rates of hole-hole and electron-electron scattering, reaches up to 1 x 1013s-' when carrier concentration is 1 x 10l8cm-3 in the wetting layers, which causes the broadening of optical gain with the full width at half maximum of 13 meV. When carrier concentration is 1 x 1015cm-3, the total scattering rate is 8 x l O ' O s - ' , giving 0.1-meV broadening. From the spontaneous emission spectra of Fig. 8.4, we see that, while carriers are almost in the ground state at 80 K, there are also carriers in the excited state and in the wetting layer at 298 K. Thus, carriers in the excited state and/or in the wetting layer work as scattering particles against electrons and holes in quantum dots, resulting in large homogeneous broadening of optical gain and one-line lasing at 298K, as seen in Fig. 8.4(b). From the viewpoint of physics, it is interesting that homogeneous broadening leads to an interaction of spatially and energetically isolated quantum dots through photons, and that collective lasing is achieved. With respect to technology, it is important that the interaction leads to a single lasing line at room temperature. This is the first demonstration of the effect of homogeneous broadening on quantum-dot laser characteristics. By further increasing currents, broad-band lasing spectra are also observed at room temperature, as seen in Fig. 8.3. This is because of the gain saturation that is due to the phonon bottleneck and/or third-order nonlinearity at high current injection, leading to an increase of the carrier population and thus lasing in the dots nonresonant to the central lasing line. For this reason, the phonon bottleneck and third-order nonlinearity dominate the purity of lasing spectra.

111. Quantum Dots on InGaAs Substrates

The growth of quantum dots on ternary InGaAs substrates is an alternative way to realize quantum-dot lasers emitting at 1.3pm. We show our preliminary results on the growth of quantum dots on InGaAs substrates. Research is underway to fabricate ternary In, -,Ga,As as a new category of substrates for optical devices besides GaAs and InP (Nakajima and Kusunoki, 1996; Kusunoki et al., 1996; Nakajima et al., 1997). We recently achieved a low threshold of 176 A/cm2 and a high characteristic temperature of 140K, a record high operating temperature up to 210"C, and temperature-insensitive slope efficiency in 1.2-pm strained quantum-well lasers on

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In,,,,Ga,,,,As substrates (Shoji et al., 1994; Shoji et al., 1996; Otsubo et al., 1997). Our aim is to fabricate 1.3-pm lasers on the ternary substrates with low lasing threshold and excellent temperature characteristics for future optical interconnection and optical subscriber systems. The merits of ternary substrates are twofold. The first is that their quantum wells have deeper potential height than the quantum wells on InP, which prevents carrier overflow into the SCH layers and results in larger optical gain and improvement of temperature characteristics (Ishikawa, 1993; Ishikawa and Suemune, 1994). Conventional strained quantum-well lasers emitting in the infrared-that is, 1.3 and 1.55pm-grown on InP substrates suffer from low temperature stability of threshold currents, primarily due to the carrier overflow from the active region to the SCH region with relatively low potential height. The carrier overflow into SCH regions with a high density of states increases the carrier number at lasing and thus accelerates the Auger recombination process, the rate of which is proportional to the cube of the carrier density, resulting in the consumption of carriers. At the same time, large carrier density at lasing greatly increases internal cavity loss, resulting in low external quantum efficiency. The second merit of ternary substrates is that high-reflectivity distributed Bragg reflector mirrors can be grown for vertical cavity surface emitting on lasers. The mirrors consisting of In,,,,Ga,~,,As and Ino,25Alo~7,As In,,,,Ga,,,,As substrates give a refractive index difference of 0.4-about twice that of the combination of InP and InGaAsP (A = 1.2pm) on InP substrates (Shoji et al., 1997). This opens the way to vertical cavity surface emitting lasers with a lasing wavelength at 1.3pm, which are the key devices in future optical interconnections. The combination of ternary substrate and self-assembled dots holds promise for 1.3-pm lasers with high-temperature stability, along with alternate-supply growth in metal organic vapor phase epitaxy (Chapter 3) and the low-growth rate method plus InGaAs covering in molecular beam epitaxy (MBE). The reason for this is that the self-assembling of quantum dots is a strain relaxation growth process, and the larger the In content in substrates, the larger the amount of In introduced in selfassembled dots. Figure 8.5(a) is an atomic field microscope image of InAs quantum dots on the (001) ,Ga,,,,As substrate by MBE. The 450-nm In,,,,Ga,,,,As buffer layer and then the dots were grown. The substrate temperature was 51OoC, the growth rate was 0.1 monolayer (ML)/sec, and 1.8-ML InAs was supplied. The dots are slightly smaller and have a larger density than dots grown on GaAs substrates. Figure 8.5(b) shows the photoluminescence spectra at 4.2K from dots grown on the (001) In,~,,Ga,,,,As substrate covered by lOOnm In,,,,Ga,~,,As, and dots grown on the (001) GaAs

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0.8

0.9

1.o 1.1 1.2 Wavelength (pm)

1.3

FIG. 8.5. (a) Atomic field microscope image of InAs quantum dots on a (001) In, ,,Ga, 8 3 A on ~ the In,, ,,Ga,,,,,As substrate. (b) Photoluminescence spectra from dots on GaAs and In,~,,Ga,,,,As substrates at 4.2K. The dots are covered by GaAs on the GaAs substrate and by In,,,,Ga,,,,,As on the In, ,,Ga,,,,As substrate.

substrate covered by 100-nm GaAs. The spectrum shifted from 1.0pm on GaAs to 1.14pm on In,,,,Ga,,,,As, which can be attributed to larger In content in the dots. The spectrum width of the dots on In,,,,Ga,,,,As is 56 meV, which is narrower than that on GaAs. Increasing the growth temperature and/or reducing the growth rate

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enhances the migration of growth atoms, resulting in larger size and longer wavelength emission. Also, by applying the close-stacking technique described in Chapter 2, room-temperature 1.3-pm emission from quantum dots can be achieved. We are now working on it.

IV. Quantum Dots Emitting at 1.3pm Grown by Low Growth Rates and with an InGaAs Cap

Using MBE, we grew self-assembled quantum dots emitting at 1.3 pm at room temperature. This was achieved by reducing the supply rate of InAs to 0.007 ML/sec, about 1/15 that in regular self-assembled dots (Chapter 2), and, at the same time, by capping the dots with 8-nm In,,,,Gao,,,As. An alternative technique for growing 1.3-pm quantum dots is to reduce the growth rate. Murray and his colleagues (1998) achieved 1.3-pm-emission quantum dots at room temperature with a very narrow spectrum width of 24meV by MBE. They explained that the thermodynamic equilibrium size at a given growth temperature is realized by the enhancement of migration length under low growth rates, The InGaAs cap of dots is also useful for increasing emission wavelength, since the cap with a lattice constant larger than GaAs releases the compressive strain of dots more than the GaAs cap does. Figure 8.6 is a sample illustration. The growth temperature was 510"C, the growth rate was 0.007 ML/sec, and the As pressure was 1 x lo-' Torr. Figure 8.7 shows the planview (a) and the cross-sectional-view (b) transmission electron microscope image. The dots have a diameter of 15 nm with an area density of 4 x 10" crnp2. The cross-sectional picture shows a dark region over Stranski-Krastanov islands, which may be due to strain field. Figure 8.8 shows the room temperature photoluminescence spectrum with

FIG. 8.6. InAs quantum dots covered with 8-nm In,,,,Ga,,,As.

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FIG.8.7. (a) Plan-view and (b) cross-sectional view transmission electron microscope image of lnAs quantum dots grown at an extremely slow rate of 0.007 ML/sec and covered with 8-nm 1% ,,GaLl 8 3 A S .

an emission wavelength of 1.3 ym from the ground state. Surprisingly, the low-density problem with the 1.3-pm dots grown by ALS is solved, showing a great possibility of lasers emitting at 1.3pm at room temperature.

V. Reduced-Temperature-inducedVariation of Spontaneous Emission in Alternate Supply (ALS) Quantum Dots Covered by In,,,Gao.,As

We found experimentally that the temperature-induced variation of the spontaneous emission wavelength is greatly reduced by covering InGaAs quantum dots with In,,,Ga,.,As instead of GaAs.

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1.o

1.1

1.2

1.3

1.4

1.5

Wavelength (pm) FIG. 8.8. Room-temperature photoluminescence spectrum of InAs quantum dots with an emission wavelength of 1.3pm from the ground state. The dots were grown at an extremely slow rate of 0.007ML/sec and covered with 8-nm In,,,,Ga,.,,As.

Figure 8.9 shows the photoluminescence emission energy from the quantum-dot ground state as a function of temperature. The samples are InGaAs quantum dots grown by ALS on GaAs substrates (Chapter 3). The dots are covered by GaAs or In,,,Ga,,,As. The inset is the emission spectrum of the dots covered by In,,,Ga,~,As at 20 K, which shows clear emission lines from the ground and second states. Unfortunately, we could not detect the emission from the Ino,,Gao,,As-covered dots above 200 K due to their low crystal quality. The emission energy decreased as temperature increased in both samples, as always observed in semiconductors. However, the amount of energy shift in the In,,,Ga,,,As-covered dots was almost half that of the GaAs-covered dots. We posit the reason for this unique phenomenon as follows. First, since the thermal expansion coefficient of In,,,Ga,.,As is about 94% that of GaAs, the temperature-induced release of compressive strain, which decreases the interband transition energy, is restricted in dots covered by In,.,Ga,.,As. Second, the asymmetric structure surrounding the dots when covered by In,.,Ga,.,As gives rise to a remarkable change in the

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’-0 *I

*--a'I

1

0

Covered by In,.,G%.,As

50

100

150 200

--a. -c

-U

250

300

-

350

Temperature (K) FIG. 8.9. Photoluminescence emission energy from the quantum-dot ground state as a function of temperature. The samples are InGaAs quantum dots grown by ALS on GaAs substrates and covered with In,,,Ga,,,As and G a s h The inset is the emission spectrum of the dots covered by InGaAs at 20 K, which shows clear emission lines from the ground and second states

dot shape as temperature increases. These two factors work to cancel the temperature-induced reduction in the bandgap of quantum-dot materials. The results tell us that the temperature-induced shift of emission energy can be artificially reduced in quantum dots, which is quite attractive for lasers with their lasing wavelength insensitive to temperature. References Ishikawa, H. (1993). A p p l . Phys. Lett. 63. 712. Ishikawa, H., and Suemune, I. (1994). I E E E Photon. Technol. Lett. 6, 344. Kusunoki, T., Nakajima, K., Shoji, H., and Suzuki, T. (1996). Mar. Res. Soc. Sj~rnp.Proc. 417, 315.

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MITSURU SUCAWARA ET AL.

Mukai, K., Nakata, Y.,Shoji, H., Sugawara, M., Otsubo, K., Yokoyama, N., and Ishlkawa, H. (1998). Electron. Lett. 34, 1588. Murray, R.,Childs, D., Malik, S., Siverns, P., Roberts, C., Hartmann, J-M., Stavrinou, P. (1998). Nakajima, K., and Kusunoki, T. (1996). J . Crysr. Growth 169, 217. Nakajima, K., Kusunoki, T., and Otsubo, K. (1997). J . Cryst. Growth 173, 42. Otsubo, K., Shoji, H., Kusunoki, T., Suzuki, T., Uchida, T., Nishijima, Y., Nakajima, K., and Ishikawa, H. (1997). Electron. Lett. 33. 1795. Shoji, H., Otsubo, K., Fujii, T., and Ishikawa, H. (1997). IEEE J . Quanfum Electron. 33, 238. Shoji, H., Otsubo, K., Kusunoki, T.. Suzuki. T., Uchida, T., and Ishikawa, H. (1996). Jpn. J . Appl. Phys. 35, L778. Shoji, H., Uchida, T., Klusunoki, T., Matsuda, M., Kurakake, H., Yamazaki, S., Nakajima, K.. and Ishikawa. H. (1994). IEEE Photon. Technol. Letr. 6, 1170.

SEMICONDUCTORS AND SEMIMETALS, VOL. 60

Index Numbers followed by the letter f indicate figures; numbers followed by the letter t indicate tables.

A

Atomic force microscopy (AFM) closely stacked InAs islands, 134, 135f InAs island growth, 121, 123-124, 123f stacked islands, 145 Atomic layer epitaxial growth, 157- 160 Atomic layer epitaxy (ALE), 156 Auger coefficient. 236-237 Auger recombination, 28, 84, 95, 332 and dephasing, 307 Auger relaxation process, 210-21 1, 213f and phonon bottleneck, 235-237 Auger scattering, 300

a.c. Stark effect, 31 Access network systems, 320 Active region, 87f in equantum-dot vertical-cavity surfaceemitting lasers, 279 in quantum dot lasers, 242, 253 Alignment of islands in-plane, 130-132 perpendicular, 125- 130 ALS quantum dots, 156 comparison to columnar SK dots, 179 crystal quality, 164, 198 diameter distribution, 187f fabrication, 156- 180 growth by InAsGaAs sequence, 160-166 growth by InGaAs sequence, 166- 176 growth process, 176-180 luminescence spectrum using NSOM, 193-194, 193f solution to low-density problem for 1.3 pm emission, 334 types, 168- 172 Anisotropy in quantum-dot vertical-cavity surfaceemitting lasers, 281 of quantum dot shape, 253 Annealing, and photoluminescence spectra, 229-23 1 Arsenic desorption, 207 Arsine (ASH,), 158 Asymmetric third-order linear susceptibility (x’~’),305, 313-314

B Band engineering in quantum well lasers, 6 Bandwidth large, and quantum dots, 303-306 of third-order linear susceptibility ( x ' ~ ) ' ) , 313 Bi-exciton lasing, 85, 98- 106 and quantum dot structure design, 102-103 Bi-exciton state, 85 Bi-excitons binding energy vs. effective mass, 98 formation, 100 formation and Stark shift, 302-303 model, 97f and population Inversion, 101 spontaneous emission and lasing, 98-106 Binding energy, 49 Bloch equation, 78-83

339

INDEX Bloch function, 5 Blue chirp, 297 Blue chirp intensity modulator, 296-298 Bohr radius, 55, 56 and density of states, 241 quantum dot radius relationship, 104 Boltzman distribution, 62 Bound-state excitons gain generation mechanisms for, 98 and population inversion, 78 Bragg reflector mirrors, 332 Broadening and indium composition, 191 and quantum dot size, 191 Broadening functions, 18, 85, 95-98 Buffer layer composition and quantum confinement control, 198-201, 199f C Capture cross section, 205 Carrier capture, 213-214 Carrier distribution characterization, 21 8 221, 219f Carrier dynamics in ALS dots, 234-235, 235f effect of thermal treatment, 229-231 and frequency rolloff, 299 and modulation response, 298-299 and quantum dot device application, 294 Carrier exchange scattering and dephasing, 307 Carrier number based on path of distribution, 220-221 from injection vs. optic excitation, 188 Carrier overflow to SCH region, 95 Carrier relaxation in discrete energy levels, 214-228 and laser performance, 86-95 model, 211-214 step-by-step process and electroluminescence simulation. 225-227 step-by-step process and photoluminescence, 224-225 study by photo/electroluminescence, 21 5218 in two-level quantum dots, 293 and wavelength chirp, 300 Carrier relaxation lifetime

effect on quantum dot lasers, 84, 86-95, 91 in quantum dots, 84 recombination lifetime relationship, 214 Carrier relaxation/recombination competition between, 23 1-235 vs. temperature, 222-224 Carrier trapping, 232 Cavity-enhanced spontaneous emission, 74 Cavity-exciton detuning, 75-76 Cavity laser, 28f, 29 with ALS quantum dots, 164 Cavity loss, 258 Cavity polariton, 66, 76-77 Cavity-polariton, 66, 77 Center-of-mass motion, 34, 41-43, 45, 55, 62 Chemical etching for narrow ridge waveguide structure, 255 Chirping. see Wavelength chirp Chirpless high-speed modulation, 295-298 Cladding layer in quantum dot lasers, 255 Closely stacked islands, 132-139 growth process, 137, 137f and size control, 132 size vs. number of layers, 134 Closely stacked quantum dot lasers, 270273 Closely stacked quantum dots, 132, 156 AnAs, 133-137 InAs/GaAs, 132-139 photoluminescence properties, 137- 139 structure, 270f Columnar quantum dot lasers, 273-276 with low-threshold current, high-output power, 325-328 performance, 328 Columnar quantum dots, 143-150, 156 Confinement, 4 controllability, 196-201 potential for quantum wells, Sf, 35, 39 quantum wells vs. closely stacked dots, 141 zero-dimensional in excitons, 140-143 Confinement potential quantum dots, 9 quantum wells, 7 quantum wire, 9 Conjugate wave function, 21 Coulomb effect

34 1

INDEX interaction among quantum dots, 85 on optical gain spectra, 77-83 Coulomb enhancement of gain, 80 Coulomb-hole self energy, 35 Coulomb potential between electron and hole, 34 Coupling constant, 23-24, 57, 60 in microcavity, 69 Creation operator, 32 Crystal quality and carrier relaxation lifetime, 91 and laser performance, 268 CW lasing, 249, 257, 267

in long-wavelength quantum dot lasers, 278 and luminescence of closely stacked quantum dots, 271 and quantum confinement control evaluation, 200-201 and zero-dimensional exciton confinement, 140-143 Differential gain in quantum dot lasers, 242 in quantum dots, 245 Differential optical gain, 92 Diirerential quantum efficiency, for quantum dot lasers, 248, 257-258 Dirac state vectors, 13 Direct modulation of quantum dot lasers, 298- 300 and wavelength chirp, 295-296 Discrete energy levels and carrier relaxation, 21 1-212 interband transitions and photoluminescence, 186- 187 and quantum dot device application, 294 in quantum dots, 215-218 Discrete subbands in quantum dot lasers, 259 Dislocation lines, 206 Distributed feedback (DFB) lasers, 295 Double exponential decay curves, 215. 219f calculated vs. measured, 22 1-225 Double heterostructure lasers, 326 Dressed excitons, 66 Dry etching for narrow ridge waveguide structure, 255 Dynamic characteristics of quantum dot lasers, 248-249

D Damping factor, 92 Decay double-exponential, 215 of time-resolved photoluminescence spectra, 217-218 Defects in InAs islands, 125 Delta function, 241, 243 Density-matrix approach and gain of quantum dots, 244-245 to linear optical susceptibility derivation, 14- 19 to nonlinear optical susceptibility derivation, 19-21 to optical susceptibility derivation, 12- 14 Density of states bulk materials, 6-7 quantum dots, 3, 5f, 10 and quantum nanostructure dimension, 6 quantum wells, 8 quantum wire. 9 Dephasing in quantum dots, 307f Dephasing rate, 12, 104 calculation, 308 -3 10 of polarization, 21 and quantum dot geometry, 311-313 for third-order linear susceptibility ( ~ ‘ ~ 307 Diamagnetic energy, 43 Diamagnetic shift of exciton optical absorption, 51 of exciton resonance in quantum wells, 43 of exciton spectra due to localization, 64-65

E

9,

Effective-mass approximation in semiconductor quantum wells. 4-7 Effective-mass equations electron-hole system, 36 excitons, 33, 38 quantum wells, 5, 39-40 quantum wells under magnetic field. 42-43 quantum wells with in-plane confinement, 40-42

342 Electric dipole approximation, 55 Electric field, 14 Electric wave function bulk materials, 108 quantum dots, 110- 1 1 1 quantum wells, 109-1 10 Electroluminescence for ALS quantum dots vs. SK dots, 188- 189 for quantum dot lasers, 255-256 simulation of spectra, 225-227 spectra, 2 15- 2 17 spectra and phonon bottleneck, 215-217 temperature dependence for quantum dot lasers, 262 Electron-hole plasma, 47 Electronic coupling of quantum dots, 250 of stacked dot layers, 272 Electronic states in quantum dot microcavity, 280f in semiconductor quantum nanostructures, 3- 1 1 Emission efficiency of ALS dots, 206 of columnar-shaped quantum dots, 274 Emission spectra broadening and island height, 130 for closely stacked quantum dots, 271 multiple-peaked for ALS quantum dots, 187-188 single island, 138f width (FWHM) for ALS quantum dots vs. temperature, 185- 186, 188f Emission wavelength and laser performance, 269 Energy-balance model for island formation, 119-121 Energy conservation rule, 25 and carrier relaxation lifetime, 210 and two-phonon processes, 212 Energy-dispersive X-ray microanalysis (EDX), 162 Energy separation and quantum dot sue, 191 Envelope wave function, 6, 25, 34 Excited states in self-assembledquantum dot lasers. 252 temporal variation, 22-23 Exciton decay

INDEX cavity effect, 67-73 in planar microcavity, 69 Exciton effect, semiconductor optical response, 12 Exciton effective-mass equations. see Effective mass equations Exciton emission spectra temporal evolution, 76 Exciton lasing conditions for, 78 in wide-gap semiconductor quantum wells, 77-78 in wide-gap semiconductors, 31-32 Exciton optical absorption spectra, 41-54 effect of magnetic field compression, 51-53, 52f as a function of temperature, 53-54, 54f Exciton-photon interactions, 43-47 control of, 66 quantum dots, 46-47 quantum wells, 45-46 semiconductor microcavities, 66-77 Exciton-polari ton dispersion relations, 66 and photon emission direction, 45 Exciton spontaneous emission cavity enhanced, 74 effect of, 31 lifetime temperature dependence, 62-63 and localization, 63-66 mesoscopic enhancement of rate, 47 in mesoscopic quantum disks, 55, 60-62 in microcavity, 68f, 75f in quantum wells, 55-60 in quantum wells and photon emission direction, 45 reversibility, 77 Excitonic enhancement of gain, 80 Excitons binding energy vs. quantum well width, 49, 50f coupling with cavity mode, 71-72 decay in planar microcavity, 69 definition, 32 effect on optical absorption spectra, 47 -54 gain generation for hound state, 98

343

INDEX integrated intensity vs. quantum well width, 49, 50f localization vs. spontaneous emission. 63-66 optical properties, 30-32 photon emission direction, 45 and population inversion, 32 radius vs. quantum well width, 49, 50f resonance in quantum wells, 31, 39, 41,43 state vectors, 32-36 temporal evolution of emission spectra, 73 External modulation scheme, 296-298

F Fabry-Perot lasers, 265,295 Fermi-Dirac distribution, 16-17 Fermi's golden rule, 57, 61, 310 Fiber probe, 193 Fiber to the curb (FTTC) network systems, 320 Fiber to the home (FTTH) network systems, 320 Four-wave mixing, 304, 304f Free excitons, 65, 70, 73-74 Frequency rollor and carrier transport, 299 Full width at half maximum (FWHM) linewidth of quantum dots, 156 for self-assembled quantum dot lasers, 25 1 and uniformity of ALS dots, 164

G

Gain properties, 47-54 quantum dot lasers, 242, 244-245 quantum dots, 258 of two-level quantum dots, 289-291 Gain saturation, 330 quantum dot lasers, 261-262, 262f semiconductor lasers, 30 two-level quantum dots, 293 and wavelength jump on columnar-dot lasers, 327 Gain spectra, 245, 245f Gaussian distributions, 42, 53, 87-88 of ALS dot diameters, 161, 187f

Gaussian inhomogeneous broadening, 8788 Ground-state emission energy, and quantum dot size, 191 Growth rate for 1.3 pm emitting quantum dots, 333-335

H

Harmonic-oscillator type quantum confinement, 189 Heisenberg equation, 79 Highly lattice-mismatched semiconductors, 118 Hole, definition, 32 Homogeneous broadening. 77.82 effect on lasing emission spectra, 95-98, 328-331 origin, 330-331 for single quantum dots, 85, 192 for single SK dots, 185

I InAs islands, 30 alignment, 131 close stacking, 133- 137 growth, 121-125 InAs quantum dots, 249 covered with In,,,,Ga, 8 3 A ~3341 . Infrared photodetector, 319 InCaAs cap, 333-335 InGaAs quantum dots, 249 Inhibited spontaneous emission, 74 Inhomogeneous broadening, 53-54,77, 85, 96, 106 for single SK dot, 186 for SK-mode islands, 128-129 for two-level quantum dots, 293 Injection current, relationship to carrier density, 246 Integrated intensity band-gap dependence in quantum wells, 49, 50f of exciton resonance, 51-53 Intensity modulation, 302-303 Interband optical transition. 11- 12. 19, 22, 186

344

INDEX

Interband optical transition (Contd.) and electroluminescence emission peaks, 215 in semiconductor quantum nanostructures. 11-30 Interband transition matrix in bulk materials, 111 and carrier relaxation, 212 in quantum dots, 112 Interdiffusion, 23 1 Internal loss, 95 of quantum dot lasers, 257-258 Intersubband carrier relaxation, 205, 21 1 in time-resolved photoluminescence, 2 13214 Intraband relaxation and relaxation broadening, 245 and width-of-gain spectrum, 243 Island growth, 118 energy balance model, 119- 121 multiple layers and perpendicular alignment, 125-130 on prepatterned substrates, 131 - 132 rates, 122 Islands height and emission spectrum broadening, 130 in-plane alignment, 130-132 as quantum dots, 156 stacking vs. size, 131

K k selection rule, 79 k - p band calculation method, 196 k p perturbation, 6, 30 and transition matrix in bulk materials, 111 Kramers-Kronig relation, 290

-

L LA phonons, 210-211 Laser operations closely stacked quantum dots, 272-273 long-wavelength quantum dot lasers, 276-279 quantum-dot vertical-cavity surfaceemitting lasers, 279-282

rate equations, 27-30 simulation with Auger process, 235-237 and steady state, 89 Lasers with 1.3 pm wavelength, 332, 333-335 active region, 87f crystal quality, 201 limiting factors to performance, 267-269 output power and Auger effect, 236-237 Lateral confinement potentials, 2 Lattice constant change and confinement control, 198-201 Lattice mismatch, 196, 198 Lattice-mismatched epitaxy, 120 Linear optical susceptibility, 14-19 LO phonons, 210-21 1 energy and thermal excitation, 307 Local area cable systems, 320 Localized exciton, 65-66 Long-haul cable systems, 320 Long-wavelength infrared lasers, 28 Long-wavelength quantum dot lasers, 276-279 Longitudinal relaxation, 13, 16 Lorentz distribution, 18, 49, 71, 87 Luttinger-Kohn parameter, 51

M Magnetic field, 51, 64 Magneto absorption, 51 Magneto-optical spectroscopy, 200-201 Many-body effect, 196 Matrix element, 44, 47 Memory duration, persistent hole burning, 315 Mesa stripes, 132 Mesoscopic enhancement, 31, 47, 104 Mesoscopic region, 77 exciton spontaneous emission in, 60-62 Metal-organic chemical vapor deposition (MOCVD), 3, 121, 242 in ALS dot growth by InGaAs sequence, 166 for self-assembled quantum dot fabrication, 250 Metal-organic vapor phase epitaxy (MOVPE), 156 and quantum dot wafer growth, 190

345

INDEX Microcavity, 55, 66-77 in quantum-dot vertical-cavity surfaceemitting lasers, 279 Microprobe photoluminescence, 192- 196 Microscope photoluminescence, 192 Misfit dislocation of superlattices, I56 Modal gain of closely stacked quantum dots, 272 current density relationship, 260 Modulation direct, 298-300 external, 296-298 intensity. 301-302 in multiple quantum well lasers, 296-298 of quantum dot lasers, 267 very high speed and quantum dots, 295-298 Modulation bandwidth, 92-93 and phonon bottleneck, 301 in quantum dot lasers, 242, 249 in SK-mode islands, 129 Molecular beam epitaxy (MBE), 3, 119. 242 growth of InAs quantum dots, 155 InAs island growth, 121- 122 self-assembled quantum dot fabrication, 250 Mott density of bi-excitons, 99, 103 Multi-mode oscillation, 265 Multiple gas injectors in MOVPE system, 166 Multiple layer growth in islands, 125-130 in SK dots vs. ALS dots, 180 in type A and type B quantum dots, 172- 176 Multiple peak emission spectra for ALS quantum dots, 187, 1 8 8 Multiple quantum well (MQW) lasers, 296-298 Multiplication factors (MPFs), 314 N

Near-field optical microscopy system (NSOM), 193, 193f ALS spectra dependence on excitation power, 194-196 Nondegenerate four wave mixing, 21 Nonexponential decay curves of timeresolved photoluminescence, 217

Nonlinear coefficient gain, 21 Nonlinear optical susceptibility, 19-21 Nonradiative centers, 206t Nonradiative channel, 231-232 Nonradiative recombination, 23, 65 ALS quantum dots, 201-207 closely stacked SK-mode islands, 144 columnar-shaped quantum dots, 148 and laser performance, 242 rate for closely stacked quantum dots, 273 Normalized broadening function, 88 0

Optical absorption spectra caused by excitons, 47-54 intensity experiment, 50-53 from linear susceptibility, 19 resonance spectrum profile, 53-54 Optical confinement factor, 28, 89 Optical data storage, 320 Optical gain bulk semiconductor lasers, 29 closely stacked quantum dots, 271 Coulomb effect on spectra, 77-83, Slf, 83f effect of homogeneous broadening, 32833 1 homogeneous broadening vs. inhomogeneous broadening in single-dots, 329 from linear susceptibility, 19 from nonlinear susceptibility, 21 quantum dots, 96 relationship with current density, 247 semiconductors, 27 Optical image processing, 320 Optical interconnections, 320 Optical mode distribution, 26 Optical nonlinearity, 2 applications, 19 Optical response categories for semiconductors, 12 Optical susceptibility, 12-14 linear, 14-19 nonlinear, 19-21 Optical switches, 2 Optically-coupled quantum dots, 250 Optoelectronics, 320 Oscillation, 76-77

346

INDEX

Oscillator strength. 98. 106 effect on quantum dot lasers, 104 Output power and carrier relaxation lifetime in quantum dot lasers, 91 columnar-shaped quantum dot lasers. 274,325-328 vs. current in columnar-dot lasers, 326f quantum dot lasers, 90f semiconductor lasers. 29

P Pauli blocking, 188- 189 Pauli's exclusion principle, 88 Periodic boundary condition, 40 Perpendicular stacking, 119 Persistent hole burning memory, 3 14-319, 316f, 317f calculated vs. measured results, 318f Phonon bottleneck, 84, 189, 209-238, 298, 300,301 description, 17, 209 and internal quantum efficiency of quantum dot lasers, 258 and laser performance, 269 Phonon scattering, 17, 49, 54, 57 Photoluminescence after annealing, 229-231 ALS quantum dots, 277f ALS quantum dots vs. SK dots, 185- 186 closely stacked islands, 137-138 closely stacked quantum dots, 271f columnar-shaped quantum dots, 148,273 columnar-shaped quantum dots vs. SK dots, 2741 and discrete state interband transitions, 186- 187 excitation in ALS quantum dots vs. SK dots, 187 InAs quantum dots with 1.3 pm emission, 335f island growth, 128-129, 130f linewidth of quantum type A vs. type B dots, 171 and nonradiative recombination channel evaluation, 201-207 quantum dots, 10, 1If quantum wells, 63, 65

reduction of temperature induced shift, 336-337 single island layer, 138f SK dots vs. excitation power, 214f time-resolved, 217-225 TMIDMEA grown ALS dots, 160, 161f wavelength tunability in ALS dots, 164 wavelength vs. ALS dot growth temperature, 168, 169f wavelength vs. quantum dot size, 164 Photon lifetime, 22, 74-75 Photons in quantum dot microcavity, 280f reabsorption in microcavity, 71 wave vector, 72 Plasma dispersion in quantum dots, 290 Polarization, 14, 16, 80 bandwidth and dephasing rate, 21 and interband optical transition, 14 of quantum-dot vertical-cavity surfaceemitting lasers, 281 Population inversion, 27 in bi-excitons, 99, 101 in bound-state excitons, 78 in excitons, 32 Pulse jet epitaxy (PJE), 157 and ALS dot growth by InAsGaAs sequence, 160 and laser devices. 164

Q Quantized energies, controlling, 196-200 Quantum-box laser, 241 Quantum-confined Stark effect, 2, 31, 95 Quantum confinement, 118, 156 controllability, 196-201 effect on exciton radius, 49 harmonic oscillator type, 189 potential evaluation by diamagnetic shifts, 200-201 Quantum disks, 32, 55 mesoscopic and spontaneous photon emission, 77 spontaneous photon emission in microcavity. 66-77 Quantum-dot infrared photodetector, 294 Quantum dot injection layers, 243 Quantum dot lasers, 83-107 characteristics, 255-267

347

INDEX Coulomb effect, 84 direct modulation, 298-300 efficency and carrier relaxation lifetime, 91 emerging technologies, 269-282 emission efficiency, 257 on GaAs substrates, 95 hierarchy, 94, 94f on I n P substrates, 95 lasing characteristics, 256-262 mirror loss dependence, 257-258 output vs. current, 25% properties, 243-249 schematic, 246f self-assembly process, 84 spectral linewidth, 242, 249 speed of carrier relaxation, 84 temperature dependence of characteristics, 262-267 wavelength chirp, 300, 301f Quantum dot layers, optical coupling, 250 Quantum dot optical devices, 320-321 Quantum dot size and excited states: 252 and laser performance, 268 uniformity, 251 Quantum dots approaches to fabrication, 242 atomic layer epitaxial growth, 152- 160 bleaching, 316 and Coulomb interactions, 85 density and laser performance, 268 density in SK dots, 250 density of states, 10 direct modulation, 295-296 electroluminescence properties, 185- 189 emitting at 1.3 pm wavelength, 333-335 fabrication, 118 on InGaAs substrates, 331-333 intensity modulation, 302-303 interaction and homogeneous broadening, 33 1 light emission from discrete energy states, 185-196 as nonlinear medium, 303-314 optical characterization, 183-207 optical nonlinearity, 303-306 optimization of structure, 250 photoluminescence excitation, 185- I89 photoluminescence properties, 185- 189

photoluminescence spectra, 10, 1 1 f properties, 288-294 selective growth on patterned substrate, 242 self-assembly process, 242 single dot optical properties, 85 size control by close stacking, 139 third-order nonlinear susceptibility, 291292 as a two-level system, 288-293 uniformity and laser performance, 268 and very high speed light modulation, 295-298 wave function, 10 Quantum nano-structures density of states, 4, 4f optical response, 1 1-30 Quantum well infrared photodetector (QWIP), 319 Quantum-well lasers, 2 wavelength chirp, 300, 301f Quantum well thickness comparison t o SK mode wetting layer, 171, 176 Quantum wells, 2, 7-8 energy calculation by effective-mass approximation, 4-7 spontaneous emission. 55-66 Quantum wire, 140- 143 density of states, 9 fabrication, 1 1 8 wave function, 9 Quantum-wire lasers, 300. 301f

R Rabi frequency, 72 Radiative emission efficiency of ALS dots, 201-207 Radiative lifetime, 247 Random carrier capturing, 21 5--216 Random initial occupation (RIO), 217-21 8 and decay curves of time-resolved photoluminescence, 225 Rate equations carrier-photon system, 87 laser operations, 27-30 photons, 292 Reading signal, 315, 317f Reciprocal space map, 198

INDEX Recombination and dephasing, 307 Recombination lifetimes for ALS quantum dots, 203, 203f Red chirp, 297 in quantum dots, 290 Reduced effective mass, 40 Reflection high-energy electron diffraction (RHEED) InAs closely stacked islands, 133, 134f InAs island growth, 121, 122-123. 122f stacked island growth, 144, 145f Refractive index dispersion and external modulation, 297 and quantum dot device application, 294 quantum well, 290f Relative motion, 35-36, 41, 43 Relaxation oscillation frequency, 92 quantum dot lasers, 248 Resonant excitation, 21 1-212 Resonant modes in microcavity, 280 Retarded carrier relaxation. see Phonon bottleneck Ridge-waveguide structure, 253-254, 253f Rolloff, 92, 94, 299

S Saha equation, 100 Schrodinger equation, 56 time-dependent, 13, 22 Second quantization, 32,43 Selection rule, for wave vectors, 43 Self-assembled growth, 119 Self-assembled quantum dot lasers electroluminescence properties, 255 -256 fabrication, 249-255 modulation characteristics, 267 optical properties, 251 -253 temperature dependence, 262-267 Self-assembled quantum dots, 2 alternative supply in atomic layer epitaxy (ALE), 242 fabrication, 253-255 fabrication by SK mode, 250 grown by metal organic chemical vapor deposition (MOCVD), 155-181 InAsGaAs sequence growth. 161 molecular beam epitaxial growth, 117151

self-organization on disk structures, 242 spontaneous formation in SK mode, 242 structure of ALS type, 163 Self-generated electro-optic effect devices (SEEDS), 2, 31 Self-limiting growth for ALS quantum dots, 158-159, 167 Semiconductor Bloch equation, 78-83 Semiconductor lasers edge-emitting type, 27 effect of nonlinearity, 19 principle of operation, 27 Semiconductor quantum nano-structures electronic states, 3-1 1 fabrication, 118 history of development, 1-3 interband optical transition, 11-30 optical susceptibility, 12-21 spontaneous emission rate, 21-26 Semiconductor superlattice, 2 Separate confinement heterostructure (SCH) layer, 86 in columnar-shaped quantum dot lasers, 326 Shottkey contact structure, 316 Single quantum dot homogeneous broadening of optical gain, 85, 328-331 optical properties, 85 SK-mode islands, 128- 129, 133 Sommerfeld factor, 48, 82 Source materials for ALS quantum dots, 158 amount and ALS dot growth, 71 role in ALS dot growth, 179 role in columnar SK dots, 179 Spectral hole-burning effect, 305 Spectral hole shape, 31 8-319 Spin, 3 10 in bi-excitons, 98 in carrier number rate equations, 226 and optical susceptibility, 17 in quantum dots, 102 Spontaneous emission rate for electron-hole transition, 21-26 wavelength variation, 336-337 Spontaneous emission of bi-excitons, 98106 Spontaneous emission of photons, 21-26 cavity-enhanced, 74

INDEX and exciton localization, 63-66 inhibited, 74 lifetime variation in quantum disks vs. quantum wells, 55-62 in mesoscopic disks, 55-66 in quantum dot microcavity, 66-77 in quantum wells, 55-66 reversible, 77 temperature dependence, 62-63 Stacked islands, 148-150 Stark effect, 31 Stark shift, 302-303 State vectors, 32-36 Stimulated emission, 12, 27 Strain, 196 Strain energy, 196, 198 Strain fields and island perpendicular alignment, 127-128 Strain relaxation, 198 Strained-layer epitaxial growth, 121 Strained quantum-well laser design, 30 modulation, 95 Stranski-Krastanov (SK) mode, 3-4, 87, 155 energy-balance model for island formation, 119-121 InAs island growth, 121-125 island growth, 120- 121 multiple layer growth, 125- 130 and quantum dot growth, 118-119 for self-assembled quantum dot fabrication, 250 Strong coupling, 66-67, 71, 74 Superlattices, 156 Supply cycle number and quantum confinement control, 196- 198, 199f Surface density, 294 Susceptibility. see Optical susceptibility

T Temperature dependence ALS emission spectra width, 185- 186, 188f ALS quantum dot growth, 159, 160f, 171 ALS self-limiting growth, 167 InAs island growth, 124 photoluminescence wavelength for ALS dots, 168, 169f

quantum dot lasing characteristics, 262267 recombination lifetimes, 229-235, 233f relaxation lifetimes, 229-235. 233f Ternary substrates, 332 Thermal broadening, 53-54 Thermal excitation and dephasing. 307 Thermal treatment and phonon bottleneck, 229-235 Thermal velocity of intersubband carrier, 205 Third-order linear susceptibility (I[.’’),20, 305 amplitude, 3 I3 asymmetry, 305, 313-314 bandwidth, 313 calculation, 306-31 I multiple quantum wells, 306f and quantum dot device application, 294 two-level quantum dots, 29 1-292 Third-order nonlinear coefficient, 20--21, 30, 89 Three-dimensional quantum confinement. 241 Threshold current columnar-shaped quantum dot lasers. 273-274, 325-328 quantum dot lasers, 242, 246-248 quantum-dot vertical-cavity surfaceemitting lasers, 279-280 semiconductor lasers, 30 Threshold current density, 256 Threshold current in quantum dot lasers and carrier relaxation lifetime, 9 1 temperature dependence, 265-267, 265f Threshold gain, 29, 89 quantum dot lasers, 247 Time-dependent Schrodinger equation, 13, 22,210 Time division multiplexing (TDM), 320 Time-resolved photoluminescence, 217-225 carrier capture/relaxation process. 21 3214 T M G a (trimethylgallium), 158 TMIDMEA, 157, 158 TMIn (trimethylindium), 158 Transition matrix, 4f Transmission electron microscopy (TEM) closely stacked InAs islands, 270f

350

INDEX

Transmission electron microscopy (Conrtf.) closely stacked islands structure, 135136, 136f columnar-shaped quantum dots, 146. 147f, 274f InAs island growth, 121, 124-125, 126f InAs slow growth rate quantum dots, 335f quantum dots, 4f quantum wells, 4f triple-stacked quantum dots, 251f type A quantum dots, 171, 172f Transverse relaxation, 13, 16 Tunneling, 212 Two-band model for Bloch equation derivation, 79 for linear optical transition, 14 for nonlinear optical transition, 19 Type A quantum dots, 168-171 growth process vs. SK islands, 176 multiple layer growth, 172- 176 Type B quantum dots, 168-171 multiple layer growth, 172- 176 U

Undersea cable systems, 320

V Vector potential, 13 Vent-and-run configuration, 167 Vertical-cavity surface-emitting lasers (VCSELs), 27, 156, 279-282 with 1.3 pm wavelength, 332 schematic, 281f Vertically-stacked dots and dot density, 250 W

Wafer mapping, 190-192

Wave function bulk materials, 6 exciton-photon system, 56 exciton state, 33 normalized, of quantum wells, 7 quantum dots, 10 quantum wells, 7 quantum wire, 9 Wave vector, 22, 35, 66 optical mode, 13, 14 and selection rule for center-of-mass motion, 45 and selection rule for quantum well in-plane component, 55 Waveguide structure, 248 Wavelength behaviors in quantum dot lasers, 259f jump in columnar-dot lasers, 327 in long-wavelength quantum dot lasers, 279 Wavelength chirp, 93, 95, 291 in quantum dot lasers, 249 in quantum nano structures, 300, 301f Wavelength conversion. 12, 304 efficiency, 305f Wavelength division multiplexing (WDM), 304, 320 Weak coupling, 67, 71, 74 Wetting layer, 4. 250 absence in ALS dots, 163, 278 in columnar-shaped quantum dots, 179 comparison to quantum well thickness, 171. 176 SK islands vs. ALS dots, 161, 171 and strain energy release, 155 Wide-gap lasers, 31 -32

Z Zeeman term, 43 Zero-dimensional confinement, 140- 143

Contents of Volumes in This Series

Volume 1 Physics of 111-V Compounds C. Hilsuni, Some Key Features of 111-V Compounds F . Bassani, Methods of Band Calculations Applicable to 111-V Compounds E . 0. Kune, The k - p Method V; L. Bunch-Brueuich, Effect of Heavy Doping on the Semiconductor Band Structure D. Long, Energy Band Structures of Mixed Crystals of 111-V Compounds L. M . Roth and P. N. Argyres, Magnetic Quantum Effects S. M. Puri und T. H. Gebulle, Thermomagnetic Effects in the Quantum Region W. M. Becker, Band Characteristics near Principal Minima from Magnetoresistance E. H . Putley, Freeze-Out Effects, Hot Electron Ellects, and Submillimeter Photoconductivity

in InSb H . Weiss, Magnetoresistance B. Ancker-Johnson, Plasma in Semiconductors and Semimetals

Volume 2 Physics of 111-V Compounds M. G. Hollund, Thermal Conductivity

S. I. Novkovu, Thermal Expansion U. Piesbergen, Heat Capacity and Debye Temperatures G. Giesecke, Lattice Constants J, R. Dmhhle, Elastic Properties A. Li. Mac Rue und G. W Gobeli, Low Energy Electron Diffraction Studies R. Lee Mieher, Nuclear Magnetic Resonance B. Goldstein, Electron Paramagnetic Resonance T. S. Moss, Photoconduction in I l l - V Compounds E. Antoncik und J. Trruc, Quantum Efficiency of the Internal Photoelectric Effect in InSb G. W. Gobeli and I. G. Allen, Photoelectric Threshold and Work Function P. S. Pershan, Nonlinear Optics in 111-V Compounds M . Gershenzon, Radiative Recombination in the 111-V Compounds F. Stern, Stimulated Emission in Semiconductors

351

352

CONTENTS OF VOLUMES IN THISSERIES

Volume 3 Optical of Properties 111-V Compounds M. Huss, Lattice Reflection W. G. Spitzer, Multiphonon Lattice Absorption D. L. Stierwalt and R. F. Potter, Emittance Studies H. R. Philipp and H. Ehrenveich, Ultraviolet Optical Properties M. Cardonu, Optical Absorption above the Fundamental Edge E. J. Johnson, Absorption near the Fundamental Edge J. 0. Dimmock, Introduction to the Theory of Exciton States in Semiconductors B. Lux and J. G. Mavroides, Interband Magnetooptical Effects H. Y. Fan, Effects of Free Carries on Optical Properties E. D. Palik and G. B. Wright, Free-Carrier Magnetooptical Effects R. H. Buhe, Photoelectronic Analysis B. 0. Seruphin and H. E. Bennett, Optical Constants

Volume 4 Physics of 111-V Compounds N. A. Goryunova. A . S. Borschevskii, and D. N. Tretiakov, Hardness N. N. Sirota, Heats of Formation and Temperatures and Heats of Fusion of Compounds A"'BV D. L. Kendall, Diffusion A. G. Chynoweth, Charge Multiplication Phenomena R. W. Keyes, The Effects of Hydrostatic Pressure on the Properties of 111-V Semiconductors L. W. Aukerman, Radiation Effects N. A. Goryunova, F. P. Kesamanly, and D. N . Nasledov, Phenomena in Solid Solutions R. T. Bate, Electrical Properties of Nonuniform Crystals

Volume 5 Infrared Detectors H. Levinstein, Characterization of Infrared Detectors P. W Kruse, Indium Antimonide Photoconductive and Photoelectromagnetic Detectors M. B. Prince, Narrowband Self-Filtering Detectors I. Melngalis and T. C. Harman, Single-Crystal Lead-Tin Chalcogenides D. Long and J. L. Schmidt, Mercury-Cadmium Telluride and Closely Related Alloys E. H. Putley, The Pyroelectric Detector N. B. Stevens, Radiation Thermopiles R. J. Keyes and T. M. Quist, Low Level Coherent and Incoherent Detection in the Infrared M. C. Teich, Coherent Detection in the Infrared F R. Arums, E. W. Surd, B. J. Peylon. undF. P. Pace, Infrared Heterodyne Detection with Gigahertz IF Response H. S. Sommers, Jr., Macrowave-Based Photoconductive Detector R. Sehr und R. Zuleeg, Imaging and Display

Volume 6 Injection Phenomena M. A. Lampert and R. B. Schilling, Current Injection in Solids: The Regional Approximation Method R. Williams, Injection by Internal Photoemission A . M. Barnett. Current Filament Formation

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R. Baron and J. W Mayer, Double lnjection in Semiconductors W. Ruppel, The Photoconductor-Metal Contact

Volume 7 Application and Devices Part A J. A . Copeland and S. Knight, Applications Utilizing Bulk Negative Resistance F. A. Paduvuni, The Voltage-Current Characteristics of Metal-Semiconductor Contacts P. L. Hower, W. W. Huoper, B. R. Cairns, R. D. Fairman, and D. A. Tremere, The GaAs Field-Effect Transistor M. H. White, MOS Transistors G. R. Antell, Gallium Arsenide Transistors T. L. Tansley, Heterojunction Properties

Part B T. Misawu,lMPATT Diodes H. C. Okean, Tunnel Diodes R. B. Campbell and Hung-Chi Chang, Silicon Junction Carbide Devices R. E. Enstrom, H. Kressel, and L. Krassnur, High-Temperature Power Rectifiers of GaAs, -.P,

Volume 8 Transport and Optical Phenomena R. 1. Stirn, Band Structure and Galvanomagnetic Effects in 111-V Compounds with Indirect Band Gaps R. W. Ure, Jr., Thermoelectric Effects in IIILV Compounds H. Piller, Faraday Rotation H. Barry Bebb and E. W. Williams,Photoluminescence I: Theory E. W. Williams and H. Barry Bebb, Photoluminescence 11: Gallium Arsenide

Volume 9 Modulation Techniques B. 0. Seruphin, Electroreflectance R. L. Aggarwal, Modulated Interband Magnetooptics D. F. Blossey and Paul Handler, Electroabsorption B. But=, Thermal and Wavelength Modulation Spectroscopy I. Balslev, Piezopptical Effects D. E. Aspnes and N . Bortka, Electric-Field Effects on the Dielectric Function of Semiconductors and Insulators

Volume 10 Transport Phenomena R. L. Rhode, Low-Field Electron Transport J. D. Wiley, Mobility of Holes in 111-V Compounds C. M. Wove and G. E. Stillmun, Apparent Mobility Enhancement in Inhomogeneous Crystals R. L. Petersen, The Magnetophonon Effect

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CONTENTS OF VOLUMES I N THISSERIES

Volume 11 Solar Cells H . J. Hovel, Introduction; Carrier Collection, Spectral Response, and Photocurrent; Solar Cell Electrical Characteristics; Efficiency;Thickness; Other Solar Cell Devices; Radiation Effects; Temperature and Intensity; Solar Cell Technology

Volume 12 Infrared Detectors (11) W. L. Eisemun, J. D. Merriam, and R. E Porrer, Operational Characteristics of Infrared Photodetectors P. R. Eraft, Impurity Germanium and Silicon Infrared Detectors E. H . Putley, InSb Submillimeter Photoconductive Detectors G. E. Stillman, C. M. Wolfe, and J. 0.Dimmock. Far-Infrared Photoconductivity in High Purity GaAs G. E. Stillman and C. M. Wolfe, Avalanche Photodiodes P. L. Richards, The Josephson Junction as a Detector of Microwave and Far-Infrared Radiation E. H. Putley, The Pyroelectric Detector -An Update

Volume 13 Cadmium Telluride K. Zanio, Materials Preparations; Physics; Defects; Applications

Volume 14 Lasers, Junctions, Transport N. Holonyak, Jr. and M. H . Lee, Photopumped 111-V Semiconductor Lasers H . Kressel and J . K. Butler, Heterojunction Laser Diodes A Van der Ziel, Space-Charge-Limited Solid-state Diodes P. J. Price, Monte Carlo Calculation of Electron Transport in Solids

Volume 15 Contacts, Junctions, Emitters E. L. Sharma, Ohmic Contacts to 111-V Compounds Semiconductors A. Nussbaum, The Theory of Semiconducting Junctions J. S. Escher, NEA Semiconductor Photoemitters

Volume 16 Defects, (HgCd)Se, (HgCd)Te H. Kressel, The Effect of Crystal Defects on Optoelectronic Devices C. R. Whitsett, J. G. Broerman. and C. J. Summers, Crystal Growth and Properties of Hg, _xCd,Se alloys M. H . Weiler, Magnetooptical Properties of Hgl xCd,Te Alloys P. W. Kruse and J. G. Ready, Nonlinear Optical Effects in Hg, -xCd,Te

Volume 17 CW Processing of Silicon and Other Semiconductors J. F. Gibbons, Beam Processing of Silicon A. Lietoila, R. E. Gold, J. F. Gibbons, and L. A. Chrisrel, Temperature Distributions

and Solid Phase Reaction Rates Produced by Scanning CW Beams

CONTENTS OF VOLUMES IN THISSERIES

355

A. Leitoilu und J. F. Gibbons, Applications of CW Beam Processing to Ion Implanted Crystalline Silicon N . M . Johnson, Electronic Defects in CW Transient Thermal Processed Silicon K. F. Lee, T. J. Stultz, und J. F. Gibbons, Beam Recrystallized Polycrystalline Silicon: Properties, Applications, and Techniques T. Shibutu. A. Wukitu, T. W. S i p o n . und J. F Gibbons, Metal-Silicon Reactions and Silicide Y. I. Nissim and J. F Gibbons, CW Beam Processing of Gallium Arsenide

Volume 18 Mercury Cadmium Telluride P. W. Kruse, The Emergence of (Hg, _.,Cd,)Te as a Modern Infrared Sensitive Material H. E. Hirsch, S. C. Liung, and A. G. White, Preparation of High-Purity Cadmium. Mercury, and Tellurium W. F H. Micklethwuite, The Crystal Growth of Cadmium Mercury Telluride P. E. Petersen, Auger Recombination in Mercury Cadmium Telluride R. M. Broudy und V J. Muzurczyck, (HgCd)Te Photoconductive Detectors M. B. Reine, A . K. Soud, and T. J. TredwI/, Photovoltaic Infrared Detectors M. A. Kinch, Metal-Insulator-Semiconductor Infrared Detectors

Volume 19 Deep Levels, GaAs, Alloys, Photochemistry G. F. Neurnurk und K. Kosui, Deep Levels in Wide Band-Gap 111-V Semiconductors D. C. Look, The Electrical and Photoelectronic Properties of Semi-Insulating GaAs R. F. Brebrirk, Ching-Huu Su, and Pok-Kni Liuo, Associated Solution Model for Ga-In-Sb and Hg-Cd-Te Y. Yu. Gurevich und Y. V. Pleskon, Photoelectrochemistry of Semiconductors

Volume 20 Semi-Insulating GaAs R. N . Thomas, H. M. Hobgood3 G. W. Eldridge, D. L. Burreft. T. T. Braggins. L. B. Tu, and S. K. Wung, High-Purity LEC Growth and Direct Implantation of GaAs for Monolithic Microwave Circuits C. A. Stolte, Ion Implantation and Materials for GaAs Integrated Circuits C. G. Kirkputrick. R. T. Chen. D. E. Holrnc)s, P. M. Asbeck, K. R. Elliott, R. D. Fuirrnan. und J. R. Oliver, LEC GaAs for Integrated Circuit Applications J. S. Blukemore and S. Ruhimi, Models for Mid-Gap Centers in Gallium Arsenide

Volume 21 Hydrogenated Amorphous Silicon Part A J. I. Punkove, Introduction M. Hirose, Glow Discharge; Chemical Vapor Deposition Y. Uchida, di Glow Discharge T. D. Moustukus, Sputtering I. Yumudu, Ionized-Cluster Beam Deposition B. A. Scott, Homogeneous Chemical Vapor Deposition

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CONTENTS OF VOLUMESIN THISSERIES

F. J. Kampas, Chemical Reactions in Plasma Deposition P. A. Longeway, Plasma Kinetics H. A. Weakliem, Diagnostics of Silane Glow Discharges Using Probes and Mass Spectroscopy L. Gluttman, Relation between the Atomic and the Electronic Structures A. Chenevas-Paule, Experiment Determination of Structure S. Minomura, Pressure Effects on the Local Atomic Structure D. Adler, Defects and Density of Localized States

Part B J. I. Pankove, Introduction G. D. Cody, The Optical Absorption Edge of a-Si: H N. M. Amer and W. 5.Jackson, Optical Properties of Defect States in a-Sk H P. J. Zanzucchi, The Vibrational Spectra of a-Si: H Y. Hamakawa, Electroreflectance and Electroabsorption J. S. Lannin, Raman Scattering of Amorphous Si, Ge, and Their Alloys R. A. Street, Luminescence in a-Si: H R. S. Crandall, Photoconductivity J. Taur, Time-Resolved Spectroscopy of Electronic Relaxation Processes P. E. Vanier, IR-Induced Quenching and Enhancement of Photoconductivity and Photo luminescence H. Schade, Irradiation-Induced Metastable Effects L. Ley, Photoelectron Emission Studies

Part C J. I. Pankove, Introduction J. D. Cohen, Density o f States from Junction Measurements in Hydrogenated Amorphous Silicon P. C. Taylor, Magnetic Resonance Measurements in a-Si: H K. Morigaki, Optically Detected Magnetic Resonance J. Dresner, Carrier Mobility in a-Si: H T. Tiedje, Information about band-Tail States from Time-of-Flight Experiments A. R. Moore, Diffusion Length in Undoped a-Si: H W. Beyer a i d J . Uverhqf, Doping Effects in a-Si: H H. Fritzche, Electronic Properties of Surfaces in a-Si: H C. R. Wronski, The Staebler-Wronski Effect R. J. Nernanich, Schottky Barriers on a-Si: H B. Abeles and T. Tiedj,, Amorphous Semiconductor Superlattices

Part D J. I. Pankove, Introduction D. E. Carlson, Solar Cells G. A. Swartz, Closed-Form Solution of ILV Characteristic for a a-Si: H Solar Cells I. Shimizu, Electrophotography S. Ishioka, Image Pickup Tubes

CONTENTS OF VOLUMES IN THISSERIES

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P. G. LeComher and W. E. Speur, The Development of the a-Si: H Field-Effect Transistor and Its Possible Applications D. G. Ast, a-Si: H FET-Addressed LCD Panel S. Kaneko, Solid-state Image Sensor M . Mutsuinuru, Charge-Coupled Devices M. A . Bosch, Optical Recording A . D'Amico and G. Fortunato, Ambient Sensors H. Kukimoto, Amorphous Light-Emitting Devices R. J. Phelun, Jr., Fast Detectors and Modulators J. I. Punkove, Hybrid Structures P. G. LeComher, A . E. Owen, W. E. Speor, J. Hajro, and W. K. Choi, Electronic Switching in Amorphous Silicon Junction Devices

Volume 22 Lightwave Communications Technology Part A K. Nukfljima, The Liquid-Phase Epitaxial Growth of InGaAsP W. T. Tsang, Molecular Beam Epitaxy for 111-V Compound Semiconductors G. B. Stringfellow, Organometallic Vapor-Phase Epitaxial Growth of 111- V Semiconductors G. Beuchet, Halide and Chloride Transport Vapor-Phase Deposition of InGaAsP and GaAs M. Razeghi, Low-Pressure Metallo-Organic Chemical Vapor Deposition of Ga,In, -,ASP, - y Alloys P. M . Petrqg Defects in 111-V Compound Semiconductors

Part B J. P. van der Z i d , Mode Locking of Semiconductor Lasers

K. Y. Luu und A . Yariv, High-Frequency Current Modulation of Semiconductor Injection Lasers C. H. Henry, Special Properties of Semiconductor Lasers Y. Suemutsu, K. Kishino, S. Arai, and F. Koyctrm, Dynamic Single-Mode Semiconductor Lasers with a Distributed Reflector W. T. Tsnng, The Cleaved-Coupled-Cavity (C') Laser

Part C R. J. Nelson und M. K. Duttu, Review of InGaAsP InP Laser Structures and Comparison of Their Performance N . Chinone and M. Nukumuru, Mode-Stabilized Semiconductor Lasers for 0.7-0.8- and 1.1-1.6-pm Regions Y. Horikoshi, Semiconductor Lasers with Wavelengths Exceeding 2 pm B. A . Dean and M. Dixon, The Functional Reliability of Semiconductor Lasers as Optical Transmitters R. H. Suul, T. P . Lee, and C. A. Burus, Light-Emitting Device Design C. L. Zipfel, Light-Emitting Diode-Reliability T. P. Lee and T. Li, LED-Based Multimode Lightwave Systems K Ogaivu, Semiconductor Noise-Mode Partition Noise

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CONTENTS OF VOLUMESIN THISSERIES

Part D F. Cupusso, The Physics of Avalanche Photodiodes T. P. Peursull and M. A. Polluck, Compound Semiconductor Photodiodes T. Kunedu, Silicon and Germanium Avalanche Photodiodes S. R. Forresl, Sensitivity of Avalanche Photodetector Receivers for High-Bit-Rate LongWavelength Optical Communication Systems J. C. Cumpbell, Phototransistors for Lightwave Communications

Part E S. Wung, Principles and Characteristics of Integrable Active and Passive Optical Devices S. Murgufit und A. Yuriv, Integrated Electronic and Photonic Devices T. Mukui, Y. Yumamoto, and T. Kimuru, Optical Amplification by Semiconductor Lasers

Volume 23 Pulsed Laser Processing of Semiconductors R. F. Wood, C. W. White. and R. T. Young, Laser Processing of Semiconductors: An Overview C. W. White, Segregation, Solute Trapping. and Supersaturated Alloys G. E. Jellison. Jr., Optical and Electrical Properties of Pulsed Laser-Annealed Silicon R. F. Wood and G. E. Jeifison, Jr., Melting Model of Pulsed Laser Processing R. F. Wood and I? W. Young, Jr., Nonequilibrium Solidification Following Pulsed Laser Melting D. H. Lowndes and G. E. Jellison, Jr., Time-Resolved Measurement During Pulsed Laser Irradiation of Silicon D.M. Zebner, Surface Studies of Pulsed Laser Irradiated Semiconductors D.H. Lowndes, Pulsed Beam Processing of Gallium Arsenide R. B. James, Pulsed CO, Laser Annealing of Semiconductors R. T. Young and R. I? Wood, Applications of Pulsed Laser Processing

Volume 24 Applications of Multiquantum Wells, Selective Doping, and Superlattices C. Weisbuch, Fundamental Properties of 111-V Semiconductor Two-Dimensional Quantized Structures: The Basis for Optical and Electronic Device Applications H. Morkoc and H. Unlu, Factors Affecting the Performance of (Al, Ga)As/GaAs and (Al,Ga)As/InGaAs Modulation-Doped Field-Effect Transistors: Microwave and Digital Applications N . T. Linh, Two-Dimensional Electron Gas FETs: Microwave Applications M. Ahe rt af., Ultra-High-speed HEMT integrated Circuits D. S. Chemlu, D.A. B. Miller, and P. W. Smith, Nonlinear Optical Properties of Multiple Quantum Well Structures for Optical Signal Processing I.: Cupusso, Graded-Gap and Superlattice Devices by Band-Gap Engineering W. T. Tsung, Quantum Confinement Heterostructure Semiconductor Lasers G. C. Oshourn et ul., Principles and Applications of Semiconductor Strained-Layer Superlattices

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Volume 25 Diluted Magnetic Semiconductors W. Giriar and J. K. Furdynu, Crystal Structure, Composition, and Materials Preparation of Diluted Magnetic Semiconductors W. M. Becker, Band Structure and Optical Properties of Wide-Gap Af'xMn,B,, Alloys at Zero Magnetic Field S. Oseroff and P. H. Keesom, Magnetic Properties: Macroscopic Studies T. Giebultowicz and T. M. Holden, Neutron Scattering Studies of the Magnetic Structure and Dynamics of Diluted Magnetic Semiconductors J . Kossur, Band Structure and Quantum Transport Phenomena in Narrow-Gap Diluted Magnetic Semiconductors C. Riquaux, Magnetooptical Properties of Large-Gap Diluted Magnetic Semiconductors J. A . Guj, Magnetooptical Properties of Large-Gap Diluted Magnetic Semiconductors J. Myridski, Shallow Acceptors in Diluted Magnetic Semiconductors: Splitting, Boil-off. Giant Negative Magnetoresistance A. K, Rcrmudas and R. Rudriquez, Raman Scattering in Diluted Magnetic Semiconductors P . A. Wo@ Theory of Bound Magnetic Polarons in Semimagnetic Semiconductors

Volume 26

111-V Compound Semiconductors and Semiconductor Properties of Superionic Materials

Z. Yuanxi, 111-V Compounds H. V. Winston, A . T. Hunter. H. Kiniuru, uritl R. E. Lee, InAs-Alloyed GaAs Substrates for Direct Implantation P. K . B l ~ ~ i r c i c h a and r y ~ S. Dhar, Deep Levels in I l l - V Compound Semiconductors Grown by MBE Y. Ya. Gurcvich and A . K. fvcinov-Sliir.s. Semiconductor Properties of Supersonic Materials

Volume 27 High Conducting Quasi-One-Dimensional Organic Crystals E. M . Conwell, Introduction to Highly Conducting Quasi-One-Dimensional Organic Crystals

I. A . Howard, A Reference Guide to the Conducting Quasi-One-Dimensional Organic Molecular Crystals J. P. Pouquer, Structural Instabilities E. M . Conwell, Transport Properties (7. S. Jucohsen, Optical Properties J. C. Scort, Magnetic Properties L. Zuppiroli, Irradiation Effects: Perfect Crystals and Real Crystals

Volume 28 Measurement of High-speed Signals in Solid State Devices J . Frey and D.Ioannou, Materials and Devices for High-speed and Optoelectronic Applications H . Schumarher and E. Strid, Electronic Wafer Probing Techniques D.H. Auston, Picosecond Photoconductivity: High-speed Measurements of Devices and Materials J. A. Vulrlinunis,Electro-Optic Measurement Techniques for Picosecond Materials, Devices, and Integrated Circuits. J. M. WiesenfeldandR. K. Jain, Direct Optical Probing of Integrated Circuits and High-speed Devices G. Plows, Electron-Beam Probing A. M. Weiner und R. B. Mrtrcus, Photoemissive Probing

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CONTENTS OF VOLUMES IN THISSERIES

Volume 29 Very High Speed Integrated Circuits: Gallium Arsenide LSI M. Kuzuhara and T. Nazaki, Active Layer Formation by Ion Implantation H. Hasirnoto, Focused Ion Beam Implantation Technology T. Nozaki and A. Higashisaka, Device Fabrication Process Technology M. Zno and T. Takadu, GaAs LSI Circuit Design M. Hirayama, M. Ohmori, and K. Yamasaki, GaAs LSI Fabrication and Performance

Volume 30 Very High Speed Integrated Circuits: Heterostructure H. Watanabe, T. Mimtani, and A. Usui, Fundamentals of Epitaxial Growth and Atomic Layer Epitaxy S. Hiyamizu, Characteristics of Two-Dimensional Electron Gas in IIILV Compound Heterostructures Grown by MBE T. Nakanisi, Metalorganic Vapor Phase Epitaxy for High-Quality Active Layers T. Nimura, High Electron Mobility Transistor and LSI Applications T. Sugeta and T. Zshibashi, Hetero-Bipolar Transistor and LSI Application H. Matsuedu, T. Tanaka, and M . Nakumuru. Optoelectronic Integrated Circuits

Volume 3 1 Indium Phosphide: Crystal Growth and Characterization J. P. Farges, Growth of Discoloration-free InP M. J. McColluni and G. E. Stillman, High Purity InP Grown by Hydride Vapor Phase Epitaxy T. Znada and T. Fukuda, Direct Synthesis and Growth of Indium Phosphide by the Liquid Phosphorous Encapsulated Czochralski Method 0. O h . K. Katagiri, K. Shinohara, S. Karsura, Y. Takahashi, K Kuinosho, K. Kohiro. and R. Hirano, InP Crystal Growth, Substrate Preparation and Evaluation X Tada, M. Tatsumi, M. Morioka, T. A r k i , und T. Kawase, InP Substrates: Production and Quality Control M Razeghi, LP-MOCVD Growth, Characterization, and Application of InP Material T. A. Kennedy and P. J. Lin-Chung, Stoichiometric Defects in InP

Volme 32 Strained-Layer Superlattices: Physics T. P. Pearsall, Strained-Layer Superlattices F. H. Pollack, Effects of Homogeneous Strain on the Electronic and Vibrational Levels in Semiconductors J. Y. Marzin, J. M. Gerirrd, P. Voisin, and J. A. Erum, Optical Studies of Strained 111-V Heterola yers R. People and S. A. Jackson, Structurally Induced States from Strain and Confinement M. Jaros, Microscopic Phenomena in Ordered Superlattices

Volume 33 Strained-Layer Superlattices: Materials Science and Technology R. Hull and J. C. Bean, Principles and Concepts of Strained-Layer Epitaxy W. J. Schafi P. J. Tusker, M. C. Foisj, and L. F. Eustman, Device Applications of Strained-Layer Epitaxy

CONTENTS OF VOLUMES IN THISSERIES

361

S. T. Picraux, B. L. Doyle, and J. Y. Tsao, Structure and Characterization of Strained-Layer Superlattices E. Kasper and F. Schafer, Group IV Compounds D. L. Martin, Molecular Beam Epitaxy of IV-VI Compounds Heterojunction R. L. Gunshor, L. A. Kolodziejski, A. I/. Nurmikko,and N. Otsuka, Molecular Beam Epitaxy of 11-VI Semiconductor Microstructures

Volume 34 Hydrogen in Semiconductors J. I. Pankove and N. M. Johnson, Introduction to Hydrogen in Semiconductors C. H. S a g e r , Hydrogenation Methods J. I. Pankove, Hydrogenation of Defects in Crystalline Silicon J. W . Corbett, P. Deak, U.!L Desnicu, ond S. J . Pearton, Hydrogen Passivation of Damage Centers in Semiconductors S. J. Pearton, Neutralization of Deep Levels in Silicon J. I. Pankove, Neutralization of Shallow Acceptors in Silicon N. M . Johnson, Neutralization of Donor Dopants and Formation of Hydrogen-Induced Defects in n-Type Silicon M. Stavola and S. J. Pearron, Vibrational Spectroscopy of Hydrogen-Related Defects in Silicon A. D. Marwick, Hydrogen in Semiconductors: Ion Beam Techniques C. Herring and N . M. Johnson, Hydrogen Migration and Solubility in Silicon E. E. Huller, Hydrogen-Related Phenomena in Crystalline Germanium J. Kakalios, Hydrogen Diffusion in Amorphous Silicon J. Chevalier. B. Clerjaud, and B. Pajot, Neutralization of Defects and Dopants in 111-V Semiconductors G. G. DeLeo and W. B. Fowler, Computational Studies of Hydrogen-Containing Complexes in Semiconductors R. F. Kiefl and T. L. Estle, Muonium in Semiconductors C. G. Van de W a l k , Theory of Isolated Interstitial Hydrogen and Muonium in Crystalline Semiconductors

Volume 35 Nanostructured Systems M . Reed, Introduction H. van Houten, C. W J. Beenakker, and B. J. van Wees, Quantum Point Contacts G. Timp, When Does a Wire Become an Electron Waveguide? M. Biittiker, The Quantum Hall Effects in Open Conductors W. Hansen, J. P. Korthaus, rind U. Merkr, Electrons in Laterally Periodic Nanostructures

Volume 36 The Spectroscopy of Semiconductors D. Heitnan, Spectroscopy of Semiconductors at Low Temperatures and High Magnetic Fields A. V. Nurmikko, Transient Spectroscopy by Ultrashort Laser Pulse Techniques A. K. Ramdas and S. Rodriguez, Piezospectroscopy of Semiconductors 0. J. GIemhocki and B. V. Shanabrook, Photoreflectance Spectroscopy of Microstructures D. G. Seiler, C. L. Lirtler, and M. H. Wiler, One- and Two-Photon Magneto-Optical Spectroscopy of InSb and Hg, -xCd,Te

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CONTENTS OF VOLUMESIN THISSERIES

Volume 37 The Mechanical Properties of Semiconductors A.-B. Chen, A. Sher and W. T. Yost, Elastic Constants and Related Properties of Semiconductor Compounds and Their Alloys D. R Clarke, Fracture of Silicon and Other Semiconductors H. Siethof, The Plasticity of Elemental and Compound Semiconductors S. Guruswumy, K. T. Faber and J. P. Hirfh, Mechanical Behavior of Compound Semiconductors S. Mahajun, Deformation Behavior of Compound Semiconductors J. P. Hirth, Injection of Dislocations into Strained Multilayer Structures D.Kendull, C. B. Fleddermunn. and K. J. Malloy, Critical Technologies for the Micromachining of Silicon 1. M a m b a and K. Mokuya, Processing and Semiconductor Thermoelastic Behavior

Volume 38 Imperfections in IIIlV Materials U. Scherz and M. Schefler, Density-Functional Theory of sp-Bonded Defects in III/V Semiconductors M . Kaminska and E. R. Weber, El2 Defect in GaAs D. C. Look, Defects Relevant for Compensation in Semi-Insulating GaAs R C. Newwan, Local Vibrational Mode Spectroscopy of Defects in IIIjV Compounds A. M. Hennel, Transition Metals in III/V Compounds K. J. Malloy and K. Khachaturyan, DX and Related Defects in Semiconductors V. Swaminuthan and A. S. Jordun, Dislocations in IIIjV Compounds K. W. Nauka, Deep Level Defects in the Epitaxial IIIjV Materials

Volume 39 Minority Carriers in 111-V Semiconductors: Physics and Applications N. K. Dutfa, Radiative Transitions in GaAs and Other 111-V Compounds R. K , Ahrenkiel, Minority-Carrier Lifetime in IIILV Semiconductors T. Furuta, High Field Minority Electron Transport in p-GaAs M . S. Lundstrom, Minority-Carrier Transport in 111-V Semiconductors R A. A b r m , Effects of Heavy Doping and High Excitation on the Band Structure of GaAs D. Yevick and W. Bardyszewski. An Introduction to Non-Equilibrium Many-Body Analyses of Optical Processes in 111-V Semiconductors

Volume 40 Epitaxial Microstructures E. F. Schuberr, Delta-Doping of Semiconductors: Electronic, Optical, and Structural Properties of Materials and Devices A . Gossard, M. Sundarum, and P. Hopkins, Wide Graded Potential Wells P. Petroz Direct Growth of Nanometer-Size Quantum Wire Superlattices E. Kupon, Lateral Patterning of Quantum Well Heterostructures by Growth of Nonplanar Substrates H. Temkin. D. Gershoni, and M . Punish. Optical Properties of Ga, _,In,As/lnP Quantum Wells

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Volume 4 1 High Speed Heterostructure Devices F. Capasso, F. Beltram, S. Sen. A. Puhlevi, and A. Y. Cho, Quantum Electron Devices: Physics and Applications P. Solomon, D. J. Frank, S. L. Wright, trnd F. Cmora, GaAs-Gate Semiconductor-.InsulatorSemiconductor FET M. H. Hasherni and I/. K. Mishra, Unipolar InP-Based Transistors R. Kiehl, Complementary Heterostructure FET Integrated Circuits T. Ishibashi, GaAs-Based and InP-Based Heterostructure Bipolar Transistors H. C. Liu and T. C. L. G. Sollner, High-Frequency-Tunneling Devices H. Ohnishi, T. More, M. Takatsu, K . Imuniuru. and N. Yokoyumu, Resonant-Tunneling Hot-Electron Transistors and Circuits

Volume 42 Oxygen in Silicon F. Shimura, Introduction to Oxygen in Silicon W. Lin, The Incorporation of Oxygen into Silicon Crystals T. J. Schujier and D. K. Schroder, Characterization Techniques for Oxygen in Silicon W. M. Bullis, Oxygen Concentration Measurement S. M. Hu, Intrinsic Point Defects in Silicon 8. Pajor, Some Atomic Configurations of Oxygen J. Michel and L. C. Kimerling, Electical Properties o f Oxygen in Silicon R. C. Newman and R. Jones, Diffusion of Oxygen in Silicon T. Y. Tan and W J. Tuylor, Mechanisms of Oxygen Precipitation: Some Quantitative Aspects M. Schrerns, Simulation of Oxygen Precipitation K. Simino und I. Yonenriga, Oxygen Effect on Mechanical Properties W. Bergholz, Grown-in and Process-Induced Effects F. Shimura, Intrinsic/Internal Gettering H. Tszrya, Oxygen Effect on Electronic Device Performance

Volume 43 Semiconductors for Room Temperature Nuclear Detector Applications R. B. James and T. E. Schlesinger, Introduction and Overview L. S. Darken and C. E. Cox, High-Purity Germanium Detectors A. Burger, D. Nason, L. Van den Berg. rind M. Schieher, Growth o f Mercuric Iodide X J. Boo, T. E. Schlesinger. and R. B. Jtrnies, Electrical Properties of Mercuric Iodide X. J. Brio, R. B. James. and T. E. SchIesitIger, Optical Properties of Red Mercuric Iodide M. Hage-Ali and P. Si$erf, Growth Methods of CdTe Nuclear Detector Materials M. Huge-Ali and P S f e r r , Characterization o f CdTe Nuclear Detector Materials M. Huge-Ali and P. Sifferf, CdTe Nuclear Detectors and Applications R. B. Jumes, T. E. Schlesinger. J. Lund, rind M . Scliieber, Cd, -xZn,Te Spectrometers for Gamma and X-Ray Applications D. S. McGregor. J. E. Kammeruad Gallium Arsenide Radiation Detectors and Spectrometers J. C. Lund. F. Olschner. and A. Burger, Lead Iodide M . R. Squillunte, and K . S. Shah. Other Materials: Status and Prospects V. M. Gerrish, Characterization and Quantification of Detector Performance J. S. iwunciyk and B. E. Part, Electronics for X-ray and Gamma Ray Spectrometers M. Schieher, R. B. James, and T. E. Schle.~ingt~r. Summary and Remaining Issues for Room Temperature Radiation Spectrometers

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CONTENTS OF VOLUMES IN THISSERIES

Voiume 44 11-IV BluefGreen Light Emitters: Device Physics and Epitaxial Growth J. Hun and R. L. Gunshor, MBE Growth and Electrical Properties of Wide Bandgap ZnSe-based 11-VI Semiconductors S. Fujita w d S. Fujitu, Growth and Characterization of ZnSe-based IILVI Semiconductors by MOVPE E. Ho and L. A . Kolodziejski, Gaseous Source UHV Epitaxy Technologies for Wide Bandgap 11-VI Semiconductors C. G. Van de Walle, Doping of Wide-Band-Gap 11-V1 Compounds-Theory R Cingoluni, Optical Properties of Excitons in ZnSe-Based Quantum Well Heterostructures A. Ishibashi and A . V. Nurmikko, 11-VI Diode Lasers: A Current View of Device Performance and Issues S. Guha and J. Perruzello, Defects and Degradation in Wide-Gap IILVI-based Structures and Light Emitting Devices

Volume 45 Effect of Disorder and Defects in Ion-Implanted Semiconductors: Electrical and Physiochemical Characterization H. Ryssel, Ion Implantation into Semiconductors: Historical Perspectives You-Nian Wang and Teng-Cai Ma, Electronic Stopping Power for Energetic Ions in Solids S. T. Nakagawa, Solid Effect on the Electronic Stopping of Crystalline Target and Application to Range Estimation G. Miiller, S. Kalbirzer and G. N. Greczves. Ion Beams in Amorphous Semiconductor Research J. Eoussey-Said, Sheet and Spreading Resistance Analysis of Ion Implanted and Annealed Semiconductors M . L. Polignano and G. Queirolo, Studies of the Stripping Hall EKect in Ion-Implanted Silicon J. Stoemenos. Transmission Electron Microscopy Analyses R. Nipoti and M. Servidori, Rutherford Backscattering Studies of Ion Implanted Semiconductors P. Zaumseil, X-ray Diffraction Techniques

Volume 46 Effect of Disorder and Defects in Ion-Implanted Semiconductors: Optical and Photothermal Characterization M. Fried, T. Lohner and J. Gyulai, Ellipsometric Analysis A. Seas and C. Christofides, Transmission and Reflection Spectroscopy on Ion Implanted Semiconductors A. Othonos and C. Christofides, Photoluminescence and Raman Scattering of Ion Implanted Semiconductors. Influence of Annealing C. Christojides, Photomodulated Thermoreflectance Investigation of Implanted Wafers. Annealing Kinetics of Defects U. Zamrnit, Photothermal Deflection Spectroscopy Characterization of Ion-Implanted and Annealed Silicon Films A. Mendelis, A. Budiman and M. Vargos. Photothermal Deep-Level Transient Spectroscopy of Impurities and Defects in Semiconductors R. Kalish and S. Charbonneau, Ion lmplantation into Quantum-Well Structures A. M. Myasnikov and N. N. Gerasimenko, Ion Implantation and Thermal Annealing of 111-V Compound Semiconducting Systems: Some Problems of 111-V Narrow Gap Semiconductors

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365

Volume 47 Uncooled Infrared Imaging Arrays and Systems R. G. Buser ond M. P. Tompsett, Historical Overview P . W Kruse, Principles of Uncooled Infrared Focal Plane Arrays R . A. Wood, Monolithic Silicon Microbolometer Arrays C. M . Hanson, Hybrid Pyroelectric-Ferroelectric Bolometer Arrays D. L. Polla and J . R. Choi, Monolithic Pyroelectric Bolometer Arrays N . Teronishi, Thermoelectric Uncooled lnfrared Focal Plane Arrays M . F . Tompsett, Pyroelectric Vidicon 7: W Kenny, Tunneling Infrared Sensors J . R . Vig, R. L. Filler and Y. Kim, Application of Quartz Microresonators to Uncooled Infrared Imaging Arrays P . W Kruse, Application of Uncooled Monolithic Thermoelectric Linear Arrays to Imaging Radiometers

Volume 48 High Brightness Light Emitting Diodes G. B. Stringfellow, Materials Issues in High-Brightness Light-Emitting Diodes M . G. Craford, Overview of Device issues in High-Brightness Light-Emitting Diodes F . M . Steranka, AlGaAs Red Light Emitting Diodes C. H . Chen, S. A. Stockman, M . J . Peanasky, and C . P . Kuo, OMVPE Growth of AlGaInP for High Efficiency Visible Light-Emitting Diodes F. A . Kish and R. M . Fletcher, AlGaInP Light-Emitting Diodes M . W Hodapp, Applications for High Brightness Light-Emitting Diodes I . Akasaki and H . Amano, Organometallic Vapor Epitaxy of GaN for High Brightness Blue Light Emitting Diodes S. Nakamura, Group 111-V Nitride Based Ultraviolet-Blue-Green-Yellow Light-Emitting Diodes and Laser Diodes

Volume 49 Light Emission in Silicon: from Physics to Devices D. J. Lockwood, Light Emission in Silicon G. Abstreiter, Band Gaps and Light Emission in Si/SiGe Atomic Layer Structures T. G. Brown and D. G. Hall, Radiative lsoelectronic Impurities in Silicon and Silicon-Germanium Alloys and Superlattices J. Michel, L. V. C. Assali, M.T. Morse, and L. C. Kimerling. Erbium in Silicon Y Kunemitsu, Silicon and Germanium Nanoparticles P. M. Fauchet, Porous Silicon: Photoluminescence and Electroluminescent Devices C. Delerue, G. Allan, and M.Lannoo, Theory of Radiative and Nonradiative Processes in Silicon Nanocrystallites L. Buus, Silicon Polymers and Nanocrystals

Volume 50 Gallium Nitride (GaN) J. I. Pankove and T. D. Moustakas. Introduction S. P. DenBaars and S. Keller, Metaloryanic Chemical Vapor Deposition (MOCVD) of Group 111 Nitrides W. A . Bryden and T. J. Kistenmacher, Growth of Group IIILA Nitrides by Reactive Sputtering N. Newman, Thermochemistry of IIILN Semiconductors S. J. Pearton and R. J. Shul, Etching of 111 Nitrides

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CONTENTS OF VOLUMESIN THISSERIES

S. M. Bedair, Indium-based Nitride Compounds A. Tramperf, 0.Brandt, and K. H. Ploog, Crystal Structure of Group Ill Nitrides H. Morkoc, F. Hamdani, and A. Salvador, Electronic and Optical Properties of 111-V Nitride based Quantum Wells and Superlattices K. Doverspike and J. I. Pankove, Doping in the 111-Nitrides T. Suski and P. Perlin, High Pressure Studies of Defects and Impurities in Gallium Nitride B. Monemar, Optical Properties of GaN W. R. L. Lambrecht, Band Structure of the Group 111 Nitrides N. E. Christensen and P. Perlin. Phonons and Phase Transitions in GaN S. Nakamura, Applications of LEDs and LDs I. Akasaki and H. Ammo, Lasers J. A. Cooper, Jr.. Nonvolatile Random Access Memories in Wide Bandgap Semiconductors

Volume 5 1A Identification of Defects in Semiconductors G. D. Watkins, EPR and ENDOR Studies of Defects in Semiconductors J.-M. Spaeth, Magneto-Optical and Electrical Detection of Paramagnetic Resonance in Semiconductors T. A. Kennedy and E. R. Glaser, Magnetic Resonance of Epitaxial Layers Detected by Photoluminescence K . H. Chow, B. Hitfi, and R. F. Kiej!, pSR on Muonium in Semiconductors and Its Relation to Hydrogen K. Saarinen, P. Hautojiirvi, and C. Corbel. Positron Annihilation Spectroscopy of Defects in Semiconductors R Jones and P. R. Briddon. The Ah Initio Cluster Method and the Dynamics of Defects in Semiconductors

Volume 5 1B Identification of Defects in Semiconductors G. Davies, Optical Measurements of Point Defects P. M. Mooney, Defect Identification Using Capacitance Spectroscopy M. Stavolu, Vibrational Spectroscopy of Light Element Impurities in Semiconductors P. Schwander, W. D. Rau. C. Kisielowski. M . Gribelyuk, and A. Ourmazd, Defect Processes in Semiconductors Studied at the Atomic Level by Transmission Electron Microscopy N. D. Jager and E. R. Weber, Scanning Tunneling Microscopy of Defects in Semiconductors

Volume 52 Sic Materials and Devices K. Jarrendalil and R. F Davis, Materials Properties and Characterization of SIC V. A. Dmitriev and M. G. Spencer, Sic Fabrication Technology: Growth and Doping V. Saxena and A. J. Sfeckl, Building Blocks for SIC Devices: Ohmic Contacts, Schottky Contacts, and p-n Junctions M. S. Shur, Sic Transistors C. D. Brandt. R. C. Clarke, R. R. Siergiej, J. B. Casady, A. W. Morse, S.Sriram, and A. K. Agarwal, S i c for Applications in High-Power Electronics R. J. Trew. Sic Microwave Devices

CONTENTS OF VOLUMES IN THISSERIES

367

J. Edmond, H. Kong, G. Negley. M. Leonard, K. Doverspike, W. Weeks, A. Suvorov, D. Waltz, and C. Carter. Jr., Sic-Based UV Photodiodes and Light-Emitting Diodes H . Morkag, Beyond Silicon Carbide! Ill-V Nitride-Based Heterostructures and Devices

Volume 53 Cumulative Subject and Author Index Including Tables of Contents for Volume 1-50

Volume 54 High Pressure in Semiconductor Physics I W. Paul, High Pressure in Semiconductor Physics: A Historical Overview N . E. Christensen, Electronic Structure Calculations for Semiconductors under Pressure R. J. NeimeJ and M . I. McMahon, Structural Transitions in the Group IV, 111-V and 11-VI Semiconductors Under Pressure A. R. Goni and K. Syassen, Optical Properties of Semiconductors Under Pressure P. Trautman, M. Baj, and J. M . Buranowski, Hydrostatic Pressure and Uniaxial Stress in Investigations of the EL2 Defect in GaAs M. Li nnd P. Y Yu, High-pressure Study of DX Centers Using Capacitance Techniques T. Suski, Spatial Correlations of Impurity Charges in Doped Semiconductors N. Kuroda, Pressure Effects on the Electronic Properties of Diluted Magnetic Semiconductors

Volume 55 High Pressure in Semiconductor Physics I1 D.K. Maude and J. C. Portal. Parallel Transport in Low-Dimensional Semiconductor Structures P. C. Klipstein, Tunneling Under Pressure: High-pressure Studies of Vertical Transport in Semiconductor Heterostructures E. Anastussakis and M. Cardona, Phonons, Strains, and Pressure in Semiconductors F. H. Pollak, Effects of External Uniaxial Stress on the Optical Properties of Semiconductors and Semiconductor Microstructures A. R. Adams, M. Silver. and J. Allum. Semiconductor Optoelectronic Devices S. Porowski and I. Grzegory, The Application of High Nitrogen Pressure in the Physics and Technology of 111-N Compounds M. Yousuf; Diamond Anvil Cells in High Pressure Studies of Semiconductors

Volume 56 Germanium Silicon: Physics and Materials J. C. Beun, Growth Techniques and Procedures D. E. Savage, F. Liu, V. Zielasek, und M . G. Lagally, Fundamental Crystal Growth Mechanisms R. HUN, Misfit Strain Accommodation in SiGe Heterostructures M. J. Shaw and M. Juros, Fundamental Physics of Strained Layer GeSi: Quo Vadis? F. Cerdeira, Optical Properties S. A . Ringel and P. N. Grillot, Electronic Properties and Deep Levels in Germanium-Silicon J. C. Campbell, Optoelectronics in Silicon and Germanium Silicon K. Eberl, K. Brunner. and 0. G. Schmidi. Si,_,C, and Si,-,-,Ge,C, Alloy Layers

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CONTENTS OF VOLUMESIN THISSERIES

Volume 57

Gallium Nitride (GaN) I1

R. J. Molnar, Hydride Vapor Phase Epitaxial Growth of 111-V Nitrides T. D. Moustakas, Growth of 111-V Nitrides by Molecular Beam Epitaxy Z. Liliental- Weber, Defects in Bulk GaN and Homoepitaxial Layers C. G. Van de Walk and N. M. Johnson. Hydrogen in 111-V Nitrides W. Gotz and N. M. Johnson, Characterization of Dopants and Deep Level Defects in Gallium Nitride B. Gil, Stress Effects on Optical Properties C. Kisielowski, Strain in GaN Thin Films and Heterostructures J. A. Miragliottu and D. K. Wickenden. Nonlinear Optical Properties of Gallium Nitride B. K Meyer. Magnetic Resonance Investigations on Group 111-Nitrides M. S. Shur and M. AsifKhun, GaN and AlGaN Ultraviolet Detectors C. H. Qiu, J. I. Pankove. and C. Rousington. 111-VNitride-Based X-ray Detectors

Volume 58 Nonlinear Optics in Semiconductors I A. Kost, Resonant Optical Nonlinearities in Semiconductors E. Garmire, Optical Nonlinearities in Semiconductors Enhanced by Carrier Transport D. S. Chemlu. Ultrafast Transient Nonlinear Optical Processes in Semiconductors M. Sheik-Bahae and E. W. Van Stryland Optical Nonlinearities in the Transparency Region of Bulk Semiconductors J. E. Millerd, M. Ziari, and A. Partoit, Photorefractivity in Semiconductors

Volume 59 Nonlinear Optics in Semiconductors I1 J. B. Kkurgin, Second Order Nonlinearities and Optical Rectification K. L. Hull, E. R. Thoen, and E. P. Ippen. Nonlinearities in Active Media E. Hanamura, Optical Responses of Quantum Wires/Dots and Microcavities U. Keller, Semiconductor Nonlinearities for Solid-state Laser Modelocking and Q-Switching A. Miller, Transient Grating Studies of Carrier Diffusion and Mobility in Semiconductors

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