Semiconductor Nanolaser 9781316275122

Table of contents :
Contents......Page 5
1 Introduction......Page 10
1.1 The History of Laser Minimization......Page 11
1.2 Active Materials for Nanolasers......Page 16
1.3 Fundamental Scale Limits of Lasers......Page 18
1.4 Efficiency in Nanolasers......Page 23
1.5 Laser Rate Equations......Page 24
1.6.1 Vertical Cavity Surface-emitting Lasers (VCSELs)......Page 28
1.6.2 Photonic Crystal Defect Cavity Lasers......Page 30
1.6.3 Nanowire Lasers......Page 31
1.6.4 Cavity-free Nanolasers......Page 35
1.6.5 Metal-dielectric-metal Waveguide-based Nanolasers......Page 37
1.6.6 SPASERs......Page 42
2.1 Metallo-dielectric Cavity Design......Page 45
2.2 Invariance of Optimal Metallo-dielectric Waveguide Geometry with Respect to Metal-cladding Permittivity......Page 51
2.3 Metallo-dielectric Nanolaser Fabrication......Page 57
2.4 Optical Pump Penetration Analysis......Page 60
2.5 Metallo-dielectric Nanolasers on Silicon......Page 63
2.6 Micro-photoluminescence Characterization of Nanolasers......Page 68
3 Purcell Effect and the Evaluation of Purcell and Spontaneous Emission Factors......Page 74
3.1 Gain Medium and Its Excitation......Page 76
3.2 Formulation of Purcell Effect in Semiconductor Nanolasers at Room Temperature......Page 78
3.3 Applicability of the Formulation......Page 82
3.4 Evaluation of Purcell Effect in a Semiconductor Nanolaser......Page 83
3.5 Temperature’s Effect on FP and β......Page 87
3.6 Temperature Dependence of Cavity Modes and Emission Spectra......Page 89
3.7 Temperature Dependence of Spontaneous Emission Factor......Page 93
3.8 Design for Temperature-insensitive High-β Nanolasers......Page 97
4.1 The Fundamental Promise and Challenge of Plasmonics......Page 100
4.2.1 Modes at MD Interface......Page 103
4.2.2 Amplification in Systems of One or Several MD Interfaces......Page 105
4.2.3 Amplification in Systems of Many MD Interfaces......Page 106
4.3 MDM Lasers with 2D Confinement......Page 108
4.4 Motivation for 3D Confined Coaxial Nanolasers......Page 110
4.5 Design and Fabrication of Optically Pumped Coaxial Nanolasers......Page 111
4.6 Emission Characterization of High β-factor Coaxial Nanolasers......Page 115
4.7 Emission Characterization of Unity β-factor Coaxial Nanolasers......Page 120
4.8 Rate Equation Analysis of Unity β-factor Coaxial Nanolasers......Page 121
4.9 Perspective on Plasmonic Mode Nanolasers......Page 126
5.1 Optical Mode and Radiation Pattern of Nanopatch Lasers......Page 128
5.2 Experimental Demonstration of Optically Pumped Nanopatch Laser......Page 131
5.3 Toward Low-threshold, Engineerable Radiation Pattern, and Electrical Pumping......Page 134
6 Active Medium for Semiconductor Nanolasers: MQW vs. Bulk Gain......Page 141
6.1 Current Injection in Semiconductor Nanolasers......Page 142
6.2 Optical Cavity and Material Gain Optimization......Page 144
6.3 Reservoir Model for Semiconductor Lasers......Page 147
6.4 Laser Rate-equation Analysis with the Reservoir Model......Page 149
6.5 Discussion......Page 153
7 Electrically Pumped Nanolasers......Page 155
7.1 Optical Mode Design with Realistic Geometrical Parameters......Page 158
7.2 Cylindrical Nanolasers with InP Undercut......Page 168
7.3 Cylindrical Nanolasers without InP Undercut......Page 171
7.4 Cubical Nanolasers without InP Undercut......Page 172
8.1 Simulation of Nanolasers’ Electrical and Thermal Performance......Page 177
8.1.1 Ohmic Resistance......Page 178
8.1.2 Calculation of Self-heating......Page 180
8.1.3 Simulation of Nanolaser Heat Dissipation......Page 182
8.2 Choice and Fabrication Techniques of Dielectric Material for Thermal Management......Page 186
8.3.1 Optical Performance......Page 188
8.3.2 Electrical and Thermal Performance......Page 193
8.3.3 Discussions......Page 197
8.4.1 Experimental Validation and Optical Mode Analysis......Page 198
8.4.2 Electrical and Thermal Analysis of Measured Device......Page 202
8.5 Multi-physics Design for Room-temperature Operation......Page 205
8.6 Discussions......Page 208
9.1 Dispersion Analysis for Cavity-free Nanolaser......Page 211
9.2 Effect of Surface Roughness on Light Stopping......Page 216
9.3 Design of Stoplight Nanolasers......Page 218
10.1 Background......Page 223
10.2 Strong Coupling and Condensation between Quantum-well Excitons and Cavity Photons......Page 226
10.3 Coherent Emission of Radiation by the Stimulated Scattering of Exciton-polaritons......Page 232
10.4 Electrically pumped Polariton Microlasers......Page 233
10.5 Discussions......Page 239
11.1 State of the Art for Chip-scale Integration......Page 240
11.2 Nanolasers’ Integration with a Silicon-based Platform......Page 243
11.3.1 Far-field Engineering of Metal-clad Nanocavities......Page 245
11.3.2 Coupling from Nanolasers to Waveguides On-chip......Page 248
11.3.3 Coupling from Waveguides to Nanocavities On-chip......Page 251
11.4.1 Small-signal Modulation Dynamics......Page 254
11.4.2 Large-signal Modulation Dynamics......Page 262
11.5.1 Optically Pumped Sidewall-modulated III-V/Si DFB Microlaser......Page 265
11.5.2 Electrically Pumped Sidewall-modulated III-V/Si DFB Microlaser......Page 268
11.5.3 Coupling III-V/Si Edge-emitting Lasers to Si Waveguide......Page 270
11.5.4 Perspective: Pushing the Footprint of DFBs to the Nanoscale......Page 272
11.6 Other Applications and Future Trends of Nanolasers......Page 275
A.1 Nonrelativistic QED in Free Space and in a Resonant Cavity......Page 279
A.2 Spontaneous Emission Probability in Free Space and in a Resonant Cavity......Page 282
B.1 Analysis of the Temperature-dependent Material Gain Spectrum of Bulk In0.53Ga0.47As......Page 284
B.2 Analysis of the Temperature-dependent Material Gain Spectrum of MQW InGaAsP......Page 288
C.1 Thermal Model Overview......Page 292
C.2 Ohmic (Joule) Heating Using a Simple Stack Model......Page 293
C.3 Junction Heating......Page 295
C.5 Surface Recombination Heating......Page 297
C.6 Auger Recombination Heating......Page 298
D.1 Vertical Contact Structure......Page 299
D.2 Horizontal Contact Structure......Page 304
D.3 Discussion......Page 309
References......Page 311
Index......Page 330

Citation preview

Semiconductor Nanolasers This unique resource explains the fundamental physics of semiconductor nanolasers and provides detailed insights into their design, fabrication, characterization, and applications. Topics covered range from the theoretical treatment of the underlying physics of nanoscale phenomena, such as temperature dependent quantum effects and active medium selection, to practical design aspects, including the multi-physics cavity design that extends beyond pure electromagnetic consideration, thermal management and performance optimization, and nanoscale device fabrication and characterization techniques. The authors also discuss technological applications of semiconductor nanolasers in areas such as photonic integrated circuits and sensing. Providing a comprehensive overview of the field, detailed design and analysis procedures, a thorough investigation of important applications, and insights into future trends, this is essential reading for graduate students, researchers, and professionals in optoelectronics, photonics, applied physics, nanotechnology, and materials science. Qing Gu is Assistant Professor of Electrical Engineering at the University of Texas at Dallas, where she is directing research in the Nanophotonics Laboratory. Her research interests include the experimental investigation of miniature semiconductor lasers and other nanophotonic devices, novel light-emitting materials, quantum behavior in nanostructures, and integrated photonic circuits. Yeshaiahu Fainman is Cymer Professor of Advanced Optical Technologies and Distinguished Professor in Electrical and Computer Engineering at the University of California, San Diego. He directs research in the Ultrafast and Nanoscale Optics Group. He is a fellow of the OSA, the IEEE, and SPIE.

Semiconductor Nanolasers QING GU The University of Texas at Dallas

YESHAIAHU FAINMAN University of California, San Diego

University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107110489 10.1017/9781316275122 © Cambridge University Press 2017 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2017 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalog record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Gu, Qing, 1985– | Fainman, Yeshaiahu. Semiconductor nanolasers / Qing Gu, The University of Texas at Dallas, Yeshaiahu Fainman, University of California, San Diego. Cambridge : Cambridge University Press, 2017. | Includes bibliographical references. LCCN 2016045371 | ISBN 9781107110489 LCSH: Semiconductor lasers. | Lasers. | Miniature electronic equipment. | Semiconductors. | Nanostructured materials. LCC QC689.55.S45 G8 2017 | DDC 621.36/61–dc23 LC record available at https://lccn.loc.gov/2016045371 ISBN 978-1-107-11048-9 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

1

Introduction 1.1 The History of Laser Minimization 1.2 Active Materials for Nanolasers 1.3 Fundamental Scale Limits of Lasers 1.4 Efficiency in Nanolasers 1.5 Laser Rate Equations 1.6 Nanolaser Types and Their Characteristics 1.6.1 Vertical Cavity Surface-emitting Lasers (VCSELs) 1.6.2 Photonic Crystal Defect Cavity Lasers 1.6.3 Nanowire Lasers 1.6.4 Cavity-free Nanolasers 1.6.5 Metal-dielectric-metal Waveguide-based Nanolasers 1.6.6 SPASERs

1 2 7 9 14 15 19 19 21 22 26 28 33

2

Photonic Mode Metal-dielectric-metal–based Nanolasers 2.1 Metallo-dielectric Cavity Design 2.2 Invariance of Optimal Metallo-dielectric Waveguide Geometry with Respect to Metal-cladding Permittivity 2.3 Metallo-dielectric Nanolaser Fabrication 2.4 Optical Pump Penetration Analysis 2.5 Metallo-dielectric Nanolasers on Silicon 2.6 Micro-photoluminescence Characterization of Nanolasers

36 36

3

Purcell Effect and the Evaluation of Purcell and Spontaneous Emission Factors 3.1 Gain Medium and Its Excitation 3.2 Formulation of Purcell Effect in Semiconductor Nanolasers at Room Temperature 3.3 Applicability of the Formulation 3.4 Evaluation of Purcell Effect in a Semiconductor Nanolaser 3.5 Temperature’s Effect on FP and β 3.6 Temperature Dependence of Cavity Modes and Emission Spectra 3.7 Temperature Dependence of Spontaneous Emission Factor 3.8 Design for Temperature-insensitive High-β Nanolasers

42 48 51 54 59

65 67 69 73 74 78 80 84 88

vi

Contents

4

Plasmonic Mode Metal-dielectric-metal–based Nanolasers 4.1 The Fundamental Promise and Challenge of Plasmonics 4.2 Amplification of Propagating Modes 4.2.1 Modes at MD Interface 4.2.2 Amplification in Systems of One or Several MD Interfaces 4.2.3 Amplification in Systems of Many MD Interfaces 4.3 MDM Lasers with 2D Confinement 4.4 Motivation for 3D Confined Coaxial Nanolasers 4.5 Design and Fabrication of Optically Pumped Coaxial Nanolasers 4.6 Emission Characterization of High β-factor Coaxial Nanolasers 4.7 Emission Characterization of Unity β-factor Coaxial Nanolasers 4.8 Rate Equation Analysis of Unity β-factor Coaxial Nanolasers 4.9 Perspective on Plasmonic Mode Nanolasers

91 91 94 94 96 97 99 101 102 106 111 112 117

5

Antenna-inspired Nano-patch Lasers 5.1 Optical Mode and Radiation Pattern of Nanopatch Lasers 5.2 Experimental Demonstration of Optically Pumped Nanopatch Laser 5.3 Toward Low-threshold, Engineerable Radiation Pattern, and Electrical Pumping

119 119 122

6

Active Medium for Semiconductor Nanolasers: MQW vs. Bulk Gain 6.1 Current Injection in Semiconductor Nanolasers 6.2 Optical Cavity and Material Gain Optimization 6.3 Reservoir Model for Semiconductor Lasers 6.4 Laser Rate-equation Analysis with the Reservoir Model 6.5 Discussion

132 133 135 138 140 144

7

Electrically Pumped Nanolasers 7.1 Optical Mode Design with Realistic Geometrical Parameters 7.2 Cylindrical Nanolasers with InP Undercut 7.3 Cylindrical Nanolasers without InP Undercut 7.4 Cubical Nanolasers without InP Undercut

146 149 159 162 163

8

Multi-physics Design for Nanolasers 8.1 Simulation of Nanolasers’ Electrical and Thermal Performance 8.1.1 Ohmic Resistance 8.1.2 Calculation of Self-heating 8.1.3 Simulation of Nanolaser Heat Dissipation 8.2 Choice and Fabrication Techniques of Dielectric Material for Thermal Management 8.3 Comparison of Device Performance with Different Dielectric Shield Material 8.3.1 Optical Performance

168 168 169 171 173

125

177 179 179

Contents

9

10

11

vii

8.3.2 Electrical and Thermal Performance 8.3.3 Discussions 8.4 Preliminary Experimental Validation and Analysis with Al2O3 Shield 8.4.1 Experimental Validation and Optical Mode Analysis 8.4.2 Electrical and Thermal Analysis of Measured Device 8.5 Multi-physics Design for Room-temperature Operation 8.6 Discussions

184 188 189 189 193 196 199

Cavity-free Nanolaser 9.1 Dispersion Analysis for Cavity-free Nanolaser 9.2 Effect of Surface Roughness on Light Stopping 9.3 Design of Stoplight Nanolasers

202 202 207 209

Beyond Nanolasers: Inversionless Exciton-polariton Microlaser 10.1 Background 10.2 Strong Coupling and Condensation between Quantum-well Excitons and Cavity Photons 10.3 Coherent Emission of Radiation by the Stimulated Scattering of Exciton-polaritons 10.4 Electrically pumped Polariton Microlasers 10.5 Discussions

214 214

223 224 230

Application of Nanolasers: Photonic Integrated Circuits and Other Applications

231

11.1 11.2 11.3

11.4

11.5

11.6

State of the Art for Chip-scale Integration Nanolasers’ Integration with a Silicon-based Platform Nanolasers’ Integration with Optical Waveguides 11.3.1 Far-field Engineering of Metal-clad Nanocavities 11.3.2 Coupling from Nanolasers to Waveguides On-chip 11.3.3 Coupling from Waveguides to Nanocavities On-chip High-speed Optical Communication with Nanoscale Light Sources 11.4.1 Small-signal Modulation Dynamics 11.4.2 Large-signal Modulation Dynamics Silicon-compatible Miniature Laser 11.5.1 Optically Pumped Sidewall-modulated III-V/Si DFB Microlaser 11.5.2 Electrically Pumped Sidewall-modulated III-V/Si DFB Microlaser 11.5.3 Coupling III-V/Si Edge-emitting Lasers to Si Waveguide 11.5.4 Perspective: Pushing the Footprint of DFBs to the Nanoscale Other Applications and Future Trends of Nanolasers

217

231 234 236 236 239 242 245 245 253 256 256 259 261 263 266

viii

Contents

Appendix A Spontaneous Emission in Free Space and Cavity A.1 Nonrelativistic QED in Free Space and in a Resonant Cavity A.2 Spontaneous Emission Probability in Free Space and in a Resonant Cavity Appendix B Temperature-dependent Material Gain B.1 Analysis of the Temperature-dependent Material Gain Spectrum of Bulk In0.53Ga0.47As B.2 Analysis of the Temperature-dependent Material Gain Spectrum of MQW InGaAsP Appendix C Modeling Thermal Effects in Nanolasers C.1 Thermal Model Overview C.2 Ohmic (Joule) Heating Using a Simple Stack Model C.3 Junction Heating C.4 Heterojunction Heating C.5 Surface Recombination Heating C.6 Auger Recombination Heating Appendix D Constriction Resistance and Current Crowding in Nanolasers D.1 Vertical Contact Structure D.2 Horizontal Contact Structure D.3 Discussion References Index

270 270 273 275 275 279 283 283 284 286 288 288 289 290 290 295 300 302 321

1

Introduction

The infrastructure that supports modern society, including health care, education, transportation, finance, and scientific and technological research, has become inextricably tied to the continuous progress in the ability to generate, transmit, receive, and process information. While electronic devices integrated in highly complex circuits have enabled this progress, electronic devices and circuits suffer from inherent limitations, namely resistor-capacitor (RC) time delays [1]. To this end, photonic devices and circuits are envisioned to complement and perhaps, eventually, supplant electronics as the enabling technology for continuous progress in the collection, transmission, and processing of information. In all photonic systems, an essential component is the light source. Optical sources with coherent emission of light, in the form of light amplification by stimulated emission of radiation (LASER), were first demonstrated in 1960 [2]. Like the first transistors, the first lasers were macroscopic devices, with footprints on the order of centimeters to decimeters. As the cavity size is reduced with respect to the emission wavelength, interesting physical effects, unique to electromagnetic cavities, arise. Experiments in the radio and microwave frequencies first demonstrated that the spontaneous emission rate of atoms in a cavity could be enhanced or inhibited, relative to their rate of emission in free space. The change in the spontaneous emission rate was found to depend on the geometry of the cavity as well as the orientation and spectra of the atoms [3]. In the 1990s, the first proposals and experiments were made to extend what has since become known as the Purcell effect to the optical regime. Modified spontaneous emission rates in dyes and semiconductors in optical microcavities were observed [4–7], and significant applications of the Purcell effect were reported, including diode lasers with greater modulation bandwidth [8–10], energy efficiency [11, 12], and absence of a threshold [13, 14]. While the concept of thresholdless operation continues to be a subject of debate [15, 16], the modulation and efficiency improvements enabled by wavelength-scale cavities are fairly well understood. For example, with proper design, the cavity of a subwavelength laser may be designed such that most of the spontaneous emission is channeled into the lasing mode [14, 17]. In so doing, unwanted emission into non-lasing modes is mitigated, and the below-threshold efficiency is limited only by non-radiative recombination. Since the observation of the Purcell effect in semiconductor cavities, lasing has been demonstrated in numerous wavelength and subwavelength scale structures. These structures include dielectric microdisks [18–21], photonic crystals [11, 22–25], nanowires [26, 27], nanomembranes [28–30], micro-pillars [31–33], and metal-clad nanocavities [14, 34–38].

2

Introduction

1.1

The History of Laser Minimization To complement micro- and nano-electronics, optical components have undergone a process of miniaturization over the past several decades. Thus, it is not an exaggeration to state that the maintenance and improvement of modern society is directly related to research advancements in photonic devices and circuits. Similar to transistors, the reduction of the size of lasers would enable a higher packing density of devices and lower power consumption per device. The first laser miniaturization came with the invention of solid-state laser diodes [39] in which the device size was reduced from meter to millimeter scale. The invention of vertical-cavity surface-emitting laser (VCSEL) [40] enabled even further miniaturization down to tens of micrometers. More recently, micro-scale whispering-gallery mode (WGM) lasers were achieved in micro-pillars/disks [18, 41] and micro-spheres [42]. In parallel efforts, advancements toward optical mode miniaturization came with the use of 2D photonic crystals in laser designs [22]. Although the optical mode sizes of these lasers may be on the order of the emission wavelength, the entire structure is quite large due to the many Bragg periods. On the other hand, micro-pillars and microdisks do not necessarily suffer from the same problem. However, until recently, packing a large number of micro-pillar/disk lasers with dielectric cavities in a small region was impractical, because the modes of these lasers are poorly confined to the active regions and may extend well beyond the physical boundaries of the cavity, leading to undesirable mode coupling of neighboring devices [19, 33]. Figure 1.1 outlines the progress in laser miniaturization over the past few decades. We see from the development time line of small lasers that the evolution of a new laser device usually takes 10–20 years: from new laser concept to first proof-of-concept optically pumped demonstration, then to electrically pumped, and in some cases to commercial applications. For practical and commercial insertion of lasers, the devices need to be continuous-wave (CW) current injected at room temperature, ideally with stable emission, reasonably long lifetime and particular properties that the already established types of lasers cannot offer. Figure 1.2 gives a quantitative comparison of the sizes of dielectric cavities in their miniaturized form and metal-clad cavities that were introduced in late 2000 (see Figure 1.1), along with typical cavity Q factor and lasing threshold. Each type of cavity or laser design has its advantage: for example, photonic crystal dielectric lasers with fewquantum-dot gain show extremely low threshold at cryogenic temperatures; dielectric cavities generally have higher Q factors than metal-clad cavities; and metal-clad cavities have much smaller dimensions than dielectric ones. In the context of subwavelength devices, one of the most figure-of-merit is size. The size of an optical cavity can be defined using different metrics, for example, the physical dimensions of the cavity or the size of the optical mode. If the goal of the size reduction is to increase the integration density such as in a laser array, then the cavity size should account not only for the overall physical dimensions of the resonator but also the spread of the optical mode beyond the physical boundary of the resonator. Throughout

1.1 The History of Laser Minimization

(a)

(b)

(c)

(d)

3

(e) Cds

MgF2 Ag

(f) l0 metallic l0 dielectric

Metal plasmon Metal non-plasmon Photonic crystal Microdisk VCSEL

1980 Figure 1.1

1985

1990

1995

2000 Year

2005

2010

Development time line of small lasers, from first demonstration to electrical, continuous-wave, and room-temperature operation, and in some cases to commercial applications. Also shown is a size comparison of various types of small lasers. The electron microscopy pictures are scaled to the free-space emission wavelength λ0 of each laser: (a) VCSEL, (b) microdisk laser, (c) photonic crystal laser, (d) metallic non-plasmon mode laser; (e) metallic propagating plasmon mode laser; (f) localized plasmon mode laser. The free-space wavelength scale bar of the metal-cavity–based lasers (d–f) is twice that of the dielectric lasers (a–c) to permit details to be seen. The metal-cavity lasers are typically smaller than λ0 and dramatically smaller than corresponding dielectric-cavity lasers. Reprinted from reference [43] with permission from Macmillan Publishers Ltd.

this book, we define the subwavelength cavities following this metric. A desired nanolaser, therefore, should be smaller than the free-space emitted wavelength in all three dimensions, in terms of both the device’s physical footprint and its optical mode confinement. Devices with such characteristics are essential for various practical applications including densely integrated chip-scale photonic circuits, displays, and sensors. By this token, Figure 1.2(a), which lists laser’s critical dimension and volume normalized to the free-space wavelength at which it emits, indicates that almost all dielectric laser cavities do not meet the metric because of either their large physical footprint or large mode volume. One of the smallest dielectric lasers is a microdisk laser with diameter smaller than its free-space emission wavelength [21]; however, it features a large effective mode volume Veff (defined as Veff = Va/Γ, where Va is the active region volume and Γ is the mode-gain overlap factor) because of poor mode confinement. Consequently, packing a large number of micro-pillar/disk lasers with dielectric cavities in a small region has been impractical. On the other hand, both distributed Bragg resonators and photonic crystal cavities can be designed to have very localized energy distribution and thus very small effective mode volumes. However, tens to hundreds of Bragg layers or lattice periods are required to confine the mode and to maintain high finesse, resulting in physical footprints that are many wavelengths in size. Only since metal-clad nanolasers’ inception in late 2000,

4

Introduction

(a) Dielectric cavity

Metal cavity 100 0.01 0.1 1 10 Laser critical dimension (in λ0), laser volume (in λ03)

(b) Dielectric cavity

Metal cavity

10

100

1,000

1,000

Laser cavity Q factor

(c) Dielectric cavity

Metal cavity

0.001 Figure 1.2

0.01

0.1 1 10 100 Laser threshold (μA or μW)

1,000

(a) Minimum extent of small lasers in one dimension (solid symbols) and minimum volume (open symbols) relative to the free-space wavelength λ0. (b) Cavity Q factor and (c) lasing threshold in micro-watts for CW optical pumping and micro-amperes for pulsed or CW electrical pumping. In (b) and (c), open symbols denote cryogenic temperature operation; filled symbols denote room temperature operation. In (a–c), for the dielectric cavities, diamond denotes VCSEL, square denotes microdisk; circle denotes photonic crystal; for metal cavities, upward triangle denotes metallic non-plasmon mode; downward triangle denotes metallic plasmon mode. Reprinted from reference [43] with permission from Macmillan Publishers Ltd.

achieving both subwavelength physical footprint and mode volume has been possible, thanks to metal cladding’s ability to strongly guide optical modes. Therefore, although we do not explicitly focus on metal-clad nanolasers in this book, we will inevitably see their reoccurrence throughout. Figure 1.2(a) shows us that the incorporation of metal cladding has significantly reduced the size of the laser cavity. Here, we discuss in more detail how such

1.1 The History of Laser Minimization

Figure 1.3

5

Dispersion relation at the interface of semiconductor and metal (in this case, silver), where z-axis denotes the wave propagation direction. The semiconductor’s permittivity is taken to be 12, representing silicon or InGaAsP. Horizontal dashed line represents dispersion relation in the uniform semiconductor, and vertical dashed line denotes the surface plasmon resonance energy. Reprinted from reference [47] with permission from Springer Publishing.

incorporation affects the nature of optical modes in nanolasers. Optical modes in metalclad cavities can be categorized into two regions: photonic mode and surface plasmonic polariton (SPP) mode. Furthermore, the SPP mode contains a sub-category of “spasing” mode. These modes’ locations in the dispersion curve are shown in Figure 1.3. The photonic mode is the classical laser mode seen in all micro- and larger-scale lasers and some nanolasers. In this case, the mode predominantly resides in the gain medium, and the propagation constant is comparable to that in a pure dielectric cavity (horizontal dashed line in Figure 1.3). The metal acts as a prefect reflector and its role is purely to reduce mode penetration into the surroundings in order to increase mode confinement. On the other hand, the SPP mode is a mode that propagates along the metal-dielectric interface in which the mode largely penetrates into the metal and the propagation constant is much larger than that in a dielectric cavity. Plasmonic effects have received immense attention since early 2000, being investigated as a means for nanoscale focusing and field enhancement. Even though it has always been known that the inherent challenge in plasmonic systems is the dissipation losses due to the constituent metal, the majority of studies in this area has focused on passive systems and does not account for the associated loss. A strategy to overcome this issue is to introduce optical gain in the dielectric constituent of the structures. Indeed, gain media can reduce the effective propagation loss and increase the SPP propagation length, enable transparent propagation, or even overcompensate propagation losses and lead to amplification and even stimulated generation of SPPs. Gain assisted lossless propagation in optical waveguide was first theoretically investigated by Nezhad et al. [44] and by Maier et al. [45], using semiconductor material as the gain medium. Both authors also envisioned the possibility of an SPP

6

Introduction

Figure 1.4

Mode profile (electric field amplitude) across the slab waveguide composed of (a) silver/ semiconductor/silver and (b) air/semiconductor/air. Waveguide width w is normalized to λ0/2 n with λ0 = 1550 nm and n = 3.46 for III-V material. Reprinted from reference [47] with permission from Springer Publishing.

nanolaser in the case of overcompensation, if there is enough gain to compensate not only the propagation loss but also the radiation loss. The extreme case of the SPP mode is when the mode energy is close to the vicinity of the metal’s SPP resonance (vertical dashed line in Figure 1.3), under which condition the mode’s propagation constant as well as loss rapidly increase and approach maxima. The stimulated optical amplification under this condition is termed surface plasmon amplification by stimulated emission of radiation (SPASER), and the emission action is called “spasing” [46]. To illustrate the role metal plays in confining the optical mode, let’s take a look at the fundamental mode’s profile across a metal/dielectric/metal and an air/dielectric/air slab waveguide with varying dielectric core widths, shown in Figure 1.4(a) and (b), respectively. With the waveguide width w normalized to λ0/2 n with λ0 = 1550 nm, we see that with metal cladding, the mode is well confined in the dielectric region with all widths, even down to w = 0.5. With air cladding, however, the fundamental mode is most confined when the width exactly matches the mode wavelength at w = 1, but it becomes poorly confined when the width is further decreased. In fact, the optical confinement factor for both waveguide types is similar when the dielectric core width is above λ0/2 n (i.e., w > 1), but as the core width decreases below λ0/2 n (corresponding to w = 1), the confinement factor for the air-clad waveguide decreases dramatically while that for the metal-clad waveguide increases dramatically. Therefore, one can loosely categorize modes in w > 1 waveguide to be photonic modes and modes in w < 1 waveguide to be plasmonic modes, complementary to the categorization based on the propagation constant in Figure 1.3. Because the radiative efficiency in semiconductor lasers is proportional to the product of the electronic density of states (EDOS) and the photonic density of states (PDOS), tailoring both densities of states (DOS) has played an important role in increasing laser efficiency and, consequently, reducing the device size. With electronics being the more

1.2 Active Materials for Nanolasers

Figure 1.5

(b) 3D re

2D re

0D re

1D re

Photonic density of states

Electronic density of states

(a)

7

cav

rph (ω)

3d

rph (ω)

(a) Electronic density of states (EDOS) for bulk (3D), quantum well (2D), quantum wire (1D), and quantum dot (0D) semiconductors. (b) Photonic density of states (PDOS) for a 3D cavity as the cavity size reduces (from top to bottom). The PDOS of vacuum is denoted by the parabolic dashed line in each panel. Reprinted from reference [47] with permission from Springer Publishing.

mature field, engineering EDOS of materials has led the effort: the tremendous progress in material growth and fabrication technologies have enabled dimensionality reduction from bulk semiconductor (3D) to quantum wells (2D), quantum wires (1D), and eventually to quantum dots (0D), resulting in their perspective EDOS shown in Figure 1.5(a). Such EDOS engineering allows us to tailor not only the density but also the location of the electronic states in the energy space to ensure efficient use of the electronic states. From the photonics point of view, the PDOS can be engineered by changing the cavity size: generally speaking, PDOS decreases with decreasing cavity size because fewer modes are allowed within the spectral window of gain, as illustrated by Figure 1.5(b). If the electronic states in Figure 1.5(a) can be aligned with the photonic states in Figure 1.5(b) such that the peaks exactly coincide, the most energy-efficient laser device can be realized.

1.2

Active Materials for Nanolasers As should become clear from Section 1.1, nanolasers have higher threshold gain compared to their larger-scale counterpart. This is because with typically lower Q factors, radiative loss becomes a larger part of the loss mechanism; with higher surface-to-volume ratio, surface recombination, which is a form of non-radiative recombination, also becomes a larger part of the loss mechanism. These higher losses must be compensated by material gain that is provided by the smaller-sized active material. Furthermore, self-heating − which lowers material gain − is more severe in smaller active volumes. As a result, only active medium with high material gain, namely liquid and inorganic solid-state material, can be used in nanolasers. Figure 1.6 summarizes optical gain materials that have been considered for use in nanolasers. Inorganic semiconductors (including bulk, quantum-well, and quantum dot) are by far the most widely used gain material in small lasers, and they offer the most practical means

8

Introduction

Bulk semiconductor

Epitaxial quantum dots

Organic dyes

Organic semiconductor

Colloidal quantum dots 100

101

102

103

Material gain G0 Figure 1.6

104

(cm–1)

Typical material gain available from different types of active media. Square denotes material gain under electrical pumping (open square denotes cryogenic temperature material gain), circle denotes material gain measured from amplified spontaneous emission (ASE) data with nanosecond optical pumping, triangle denotes material gain measured from ASE data with sub-picosecond optical pumping. Reprinted from reference [43] with permission from Macmillan Publishers Ltd.

of achieving electrically controlled and mass-producible devices. A great advantage of inorganic semiconductors over other types of gain media is the ease with which the material may be controllably doped for the formation of heterostructures suitable for electronic injection devices. The material gain provided by semiconductors ranges from a few hundred per cm from bulk gain, to a few thousand per cm from multiple-quantum-well (MQW) gain, and to tens of thousands per cm from quantum dot (QD) gain. For MQWand QD gain, one does need to take into account the fact that the modal gain is often much smaller than the material gain, which results from the limited mode-gain overlap in these types of gain media. Nonetheless, carefully designed materials such as high-density QD or strained MQW can be utilized to increase gain further and to balance out the small modegain overlap. The disadvantage of using inorganic semiconductor is that device fabrication usually uses top-down fabrication procedures and requires lithography and deposition steps that can be quite complex for nanoscale devices. Additionally, integration with other material platforms, such as those based on silicon, is not trivial. On the other hand, organic gain materials (including organic dye and semiconductor) as well as colloidal QD nanocrystals can be fabricated using the bottom-up fabrication approach that omits expensive lithography and etching steps. These gain materials can be prepared in the form of thin films with sub-micrometer precision by solution-based processes. In terms of the gain spectrum, instead of needing a priori knowledge of the desired

1.3 Fundamental Scale Limits of Lasers

9

spectrum and choosing semiconductor compositions accordingly as in the case of inorganic semiconductors, the spectrum can be readily tuned by targeted modifications of chemical composition, dimension, and structure. Although nanolasers based on organic dye are not amenable for chip-scale integration with contemporary electronics when they are embedded in mechanically flexible polymer, they find their significance in microfluidic- and optofluidic-based applications [48]. A significant feature of fluidic-based lasers is that the refractive index and gain spectrum of the active medium can be tuned in real time by changing the solvent or dye that passes through the device. As for organic semiconductors, even though they have the potential for realizing electrically controlled lasers, their low charge-carrier mobility has so far prevented the realization of electrically pumped organic semiconductor lasers. Because organic light emitting diodes (OLED) have attracted significant interest in research and lighting applications, it is worth exploring nanoscale OLED and even toward organic semiconductor nanolasers. The main disadvantage of organic gain material is photo-induced bleaching of fluorescence and gain under high-excitation conditions, leading one to question their long-term stability and maximal output power. In comparison with organic gain media, inorganic colloidal QDs retain many of the attractive properties of organic gain, such as solution-based processes and gain spectrum tenability, while offering better stability. Colloidal QDs typically have a radius between 1 and 4 nm, which is smaller than epitaxially grown QDs (whose typical radius is 3−10 nm). The material gain in thin films of colloidal QDs has been measured to be a few hundred per cm using sub-picosecond optical excitation, and lasing has been demonstrated over a wide range of wavelengths [49]. Although each type of gain material discussed earlier has its own attractiveness, in this book, we focus on nanolasers based on inorganic semiconductors, because it is the most technologically relevant gain material that is capable of current injection. Furthermore, it is compatible with the current fabrication processes for electronic integrated circuit (IC) infrastructure. Given that one of the driving forces behind semiconductor nanolaser research is the potential photonic integrated circuits (PICs) on a semiconductor chip for the fast growing information technology sector, much like the silicon-based electronic IC technology that has already infused into our everyday lives. Nano-PIC system is envisioned to supplement and expand and maybe eventually surpass the capability of electronic ICs. In current PICs, optical waveguides, splitters, couplers, and modulators can be made to subwavelength scale. However, current mainstream semiconductor lasers, while being the smallest compared to other types of lasers (e.g., dye, gas, and solid-state lasers), are still too large for PIC. Therefore, it is fitting to say that size reduction of semiconductor lasers will highly impact the development of PICs, future information technology, and other on-chip applications such as detection and sensing.

1.3

Fundamental Scale Limits of Lasers Before diving into specific nanolaser designs and the various applications of nanolasers, we first consider the fundamental limit on the size of a laser. A typical laser has two vital components. First, it must have a resonator (or cavity) that supports optical mode(s). An

10

Introduction

inherent size limitation associated with the resonator is that the cavity length in the mode propagation direction cannot be shorter than half of the mode wavelength in the medium. This is known as the diffraction limit. Second, a laser must have gain (or active) medium that is population inverted and supplies energy to the lasing mode(s). There needs to be enough gain medium such that population inversion can be achieved, which sets another restriction to the laser size. One exception is the exciton-polariton laser, which operates without a population inversion in the strong coupling regime. Exciton-polariton laser was first introduced in 1996 [50] and was demonstrated to operate under electrical injection in 2013 [12, 51]. While they provide an intriguing foundation to explore the quantum coherence of matter [52, 53], the sizes of these lasers are significantly larger than their emission wavelengths, typically in the 10 μm scale. We include a brief chapter (Chapter 10) to discuss the physics and development of exciton-polariton lasers, but they are not the focus of this book. With a conceptual understanding of the two principle elements of a laser, we use a FabryPerot cavity as an example to derive the size limits of a laser. Figure 1.7(a) shows a typical Fabry-Perot cavity. It is an optical resonator that consists of two mirrors with reflectivity R1 and R2, respectively, and the space between the mirrors is filled with gain medium. The longitudinal direction is the propagation direction along cavity length L, and the transverse direction is the waveguiding plane, both of which are schematically shown in Figure 1.7(a). For a cavity mode to get amplified, standing waves need to form in the longitudinal direction. Taking this direction to be the x-direction, physically, it means that a complex field amplitude, E0(x), at an arbitrary location x inside the cavity will maintain its original value after a round trip propagation. We denote the complex permittivity of the gain medium as εg ðωÞ ¼ ε0g ðωÞ þ iε00g ðωÞ and the wave vector as pffiffiffiffiffiffiffiffiffi ω k ðωÞ ¼ k0 ðωÞ εðωÞ, where k0 ðωÞ ¼ is the vacuum wave vector. The standing c wave requirement is mathematically expressed as pffiffiffiffiffiffiffiffiffiffi i2k ðωÞL R1 R2 e z E0 ð xÞ ¼ E0 ð xÞ

ð1:1Þ

Equation (1.1) can be understood by inspecting the effect of the real and imaginary parts of kz ðωÞ ¼ k 0z ðωÞ þ ik 00z ðωÞ separately. The minimum length, Llong;1 min , asserted by the real part of kz ðωÞ, is 2k 0z ðωÞL ¼ 2mπ; m ¼ 1; 2; 3;    Llong;1 min ¼

mπ mπ  k0 ðωÞ ¼ k 0z ðωÞ Reðn ðωÞÞ

ð1:2Þ

eff

with effective index defined as neff ðωÞ ≡ kz ðωÞ=k0 ðωÞ. In Equation (1.2), m is an integer, therefore the shortest possible cavity length is π=k 0z ðωÞ ¼ λeff =2 where λeff is the wavelength of light inside the cavity. Equation (1.2) describes the half-wavelength condition, also known as the diffraction limit. This applies to all electromagnetic waves and sets a lower limit to the cavity size. Similarly, the minimum length in the transverse direction is also set by the diffraction limit and is denoted by Ltrans min .

1.3 Fundamental Scale Limits of Lasers

Figure 1.7

11

(a) Schematic of a conventional Fabry-Perot cavity. The minimum length L of the laser resonator is given by a phase-matching limit and a gain limit (i.e., we must compensate for light-extraction losses at the two mirrors – with reflectivities R1 and R2 – and internal absorption losses αi). The minimum volume of the active region Va is also determined by the gain limit. Here, d is the diameter of the active material waveguide between the mirrors. (b) The total mode volume Vm, determined by the mode’s electric field profile |E|2, depends on the geometry of the resonator and the transverse dimensions of the Fabry-Perot laser. Vm can be reduced by tuning d. For a thin dielectric waveguide (left), the spatial extent DH of the transverse H field is greater than for the plasmon gap mode of a metal-insulator-metal (MIM) waveguide (right). Reprinted from reference [43] with permission from Macmillan Publishers Ltd.

On the other hand, Llong;2 min , asserted by the imaginary part of kz ðωÞ is pffiffiffiffiffiffiffiffiffiffi 2k 00ðωÞL R1 R2 e z ¼1 Llong;2 min ¼

lnðR1 R2 Þ lnðR1 R2 Þ ¼ 4kz00ðωÞ 2ðGm  αi Þ

ð1:3Þ

where Gm is the modal gain and αi is the propagation loss. Gm is typically related to the material gain Ga of the active region material through the confinement factor Γ, which

12

Introduction

describes the fractional overlap of the laser mode volume V0 and the volume of the active region Va Gm ¼ ΓGa

ð1:4Þ

Equation (1.3) describes the relationship between the length requirement and Gm vs. αi, ensuring that in each round trip, the modal gain overcomes cavity loss, including the internal absorption loss and the mirror losses. This is the underlying principle behind the design of the second component of a laser. In a conventional macro-scale Fabry-Perot cavity, Llong;2 is much larger than Llong;1 min min . As the cavity size reduces to the nanoscale, long;1 Llong;2 min approaches Lmin . We therefore raise the question: how do lasers become compatible with electronic device dimensions, which can measure as small as hundreds and even tens of nanometers? In the field of nanolasers, the ultimate goal is to reduce the characteristic long;2 trans lengths, Llong;1 min , Lmin , and Lmin , without violating any laws of physics. We first examine the length restriction on the transverse dimension, Ltrans min . The length in the transverse plane is denoted by “d” in Figure 1.7(a), describing the width of the waveguide core. Although the first order transverse modes have no cut-off frequency and therefore no cut-off size requirement, guided modes transition into radiation modes as the width of the waveguide core decreases beyond half-wavelength. This is schematically illustrated in Figure 1.7(b, left), in which a significant portion of the modal profile spreads into the dielectric cladding as d decreases to the subwavelength scale in a conventional dielectric surroundings. Such mode spreading greatly reduces the gainmode overlap and increases radiation loss. trans To reduce the diffraction limited characteristic length Llong;1 min and Lmin , one needs to   0 increase k z ðωÞ, or equivalently, Re neff , in Equation (1.2). Similar to that employed in optical fibers, the most intuitive method is to increase the absolute refractive index of the gain medium in the waveguide core while minimizing that of the dielectric surrounding. For example, for semiconductor lasers in the 1550 nm telecommunication wavelength regime, the gain medium refractive index is around 3, resulting in a half-wavelength of only 250 nm in a purely dielectric environment. To further reduce the half-wavelength, one can increase the mode confinement factor   Γ and thus Re neff , by incorporating metal into device designs. Because of the negative real part of metal permittivity at optical frequencies, field penetration into metal is minimal. As a result, a metal-clad waveguide can guide modes that would otherwise be radiating in a pure dielectric waveguide of the same subwavelength core size. This is illustrated in Figure 1.7(b, right), in which the cavity has the same dimensions and materials as that of Figure 1.7(b, left) but with the addition of metal-cladding. We see clearly that a subwavelength localized transverse field is guided. The downside of using the metal-cladding is that the material loss in metals, αi, as a result of the large positive imaginary part of the permittivity that is significant at optical frequencies. The relationship between the spatial mode extent DH and the active medium length d in the transverse plane is depicted in Figure 1.8, for both purely dielectric and

1.3 Fundamental Scale Limits of Lasers

1

13

1

αi (×1,000 cm–1)

Spatial extent DH/λ0

2

Dielectric MIM 0

0 0

0.04

0.08

0.12

0.16

d/λ0 Figure 1.8

DH against d, both normalized to the free-space emission wavelength λ0, for dielectric and MIM waveguides. The reduced DH of the MIM waveguide comes at the cost of increased αi (dashed line). Calculations assume λ0 = 1.55 μm, silver as the metal, 3.6 as the bulk refractive index for the active material and air as the surrounding dielectric. Reprinted from reference [43] with permission from Macmillan Publishers Ltd.

metal-insulator-metal (MIM) waveguides. We can observe that while the inclusion of metal-cladding does not affect the mode profile by much at large d, it greatly changes the mode profile as d becomes deeply subwavelength – a property that makes the choice of metal-cladding favorable in nanolasers. On the other hand, the material loss αi also increases with decreasing d, as shown by the dashed line in Figure 1.8. This material loss is also related to the second length limitation, Llong;2 min , determined by interplay of gain and loss in a laser. To reduce Llong;2 in Equation (1.3), the modal gain and/or the end-facet reflectivity min need to be increased. Referring to Equation (1.4), while material gain G0 is fixed for any material at a given pump level and temperature, Γ can be manipulated by cavity mode design. In conventional lasers where waveguiding is weak and the wave propagation is qausi-paraxial, Γ can be defined as either the power or the energy in the gain medium divided by that of the entire optical mode. In nanolasers, this power confinement factor definition becomes invalid, because it may exceed unity as the guiding becomes strong. Instead, Γ is defined as the energy confinement factor in nanolasers [54]. It has been proposed that large modal gain can be achieved in metal-clad nanolasers [55, 56]. In terms of facet reflectivity, although enhancing the reflectivity can shorten Llong;2 min , it also limits the available output power. Additionally, the Fresnel formula used to approximate mirror loss in conventional lasers breaks down on the nanoscale, and more accurate calculation needs to be used [57]. Laser miniaturization therefore involves the search for the optimal design of constituent materials, optical modes, cavity geometry, and end-facet reflectivity, such that the significant material and end-facet losses can be compensated by the high modal gain in a small cavity.

14

Introduction

1.4

Efficiency in Nanolasers Several figures-of-merit, such as output power, lasing threshold, quantum efficiency, and wall-plug efficiency, are typically used to determine the efficiency of lasers, and lightemitting devices in general. For nanolasers, several more parameters that are specific to nanolaser efficiency are often discussed as well. The quantum efficiency (or quantum yield) is often of interest for processes that convert light; it is defined as the percentage of the input photons that contributes to the desired effect. For semiconductor lasers, a photon is generated through the radiative recombination of an electron-hole pair. The capability of a single electron-hole pair to generate a photon is called the internal quantum efficiency, ηint. It is defined as ηint ≜ ¼

rate of radiative recombination rate of radiative recombination þ rate of non-radiative recombination 1=τr 1=τr þ 1=τnr

ð1:5Þ

where τr and τnr are the mean lifetime of radiative and non-radiative recombination of electron-hole pairs, respectively. ηint is used to characterize the internal generation of photons as its name suggests; therefore, it does not take into account how many carriers injected into the device do not excite the material to the emitting excited state. Nor does it take into account how many generated photons get lost before exiting the device, for example, by reabsorption processes. To account for all carriers injected into a light source, the external quantum efficiency, ηext is thus introduced. It is defined as ηext ≜

rate of photons exiting the light source ¼ ηj  ηext  ηx rate of carriers entering the light source

ð1:6Þ

where ηi is carrier injection efficiency and ηx is the photon extraction efficiency. Another often used benchmark for energy efficiency in light sources is the wall-plug efficiency, sometimes also called power efficiency, of a laser system, which is its total electrical-to-optical power efficiency, that is, the ratio of the light output power to the electrical input power. In strict terms, the electrical power should be measured at the wall plug, so that this efficiency includes losses in the power supply and also the power required for a cooling system, which can be significant for high-power lasers. However, it is common to loosely define the wall-plug efficiency to be the electric power delivered to the laser diodes, ignoring the actual wall plug. Although this efficiency is formally defined for electrically pumped light sources, it can also be used for optically pumped ones. Wall-plug efficiency as high as 62% has been achieved in VCSELs [58]; but some laser systems, for example, titanium-sapphire lasers pumped by argon ion lasers, have wall-plug efficiencies around or below 0.1%. All the previously mentioned efficiencies certainly apply to nanolasers. However, because nanolasers are still in the initial proof-of-concept stage, be it optically pumped lasers or electrically pumped ones, efficiency in these lasers has not been systematically

1.5 Laser Rate Equations

15

characterized. Currently, nanolaser efficiency and performance are typically evaluated based on the lasing threshold, operating temperature, as well as Purcell factor and spontaneous emission factor. Purcell effect is a phenomenon that describes the spontaneous emission rate of an emitter in a cavity compared to emission in free space, which may be enhanced or inhibited. The spontaneous emission modification factor, also known as the Purcell factor, describes the rate of cavity mode spontaneous emission relative to the emission in bulk material. Directly related to the Purcell factor is the spontaneous emission factor, which quantifies the fraction of spontaneous emission into the lasing mode compared with spontaneous emission into all modes. With all else being equal, the higher the Purcell factor is for the lasing mode, the higher the spontaneous emission factor is. Purcell effect exists in all lasers, but in large-scale semiconductor lasers the difference between Purcell factors in different lasers is minuscule and can often be ignored. As a result, the spontaneous emission factor is usually on the order of 1×10−5. In nano-scale lasers, however, enhanced emission together with a reduced number of cavity modes in a small mode volume can have significant effects, especially on subthreshold behavior. These effects are generally desirable, as they tend to increase the utilization of spontaneous emission into the lasing mode (leading to higher spontaneous emission factor) and lower the lasing threshold. We look into how the larger Purcell factor and spontaneous emission factor affect the lasing threshold in the next section and will quantitatively evaluate them in Chapter 3. Generally, among the many optical modes (including both resonant and leaky modes) in a cavity, one of the resonant modes undergoes a transition to laser oscillation, whereas the other modes generate wasteful spontaneous emission. In designing a laser with maximum spontaneous emission efficiency, one should i. Eliminate, as far as possible, the wasteful resonant modes from the spectral region of gain; ii. Minimize, as far as possible, the leaky modes into the free space; iii. Utilize the resonant mode with a sufficiency high cavity Q factor, and even more importantly, an as small as possible modal volume in order to maximize the interaction between the optical mode and the gain medium; iv. Align, as much as possible, the gain spectrum to the resonant mode in (iii).

1.5

Laser Rate Equations The rate equations are a standard and widely used method in studying both the steady state and dynamical behavior of a laser. For nanolasers, however, extra care needs to be applied. For example, the Purcell effect, which characterizes the change of spontaneous emission rate in a subwavelength cavity with respect to that in free space, needs to be incorporated into the rate equations; the normalization of the optical field needs to be calculated properly; and the dispersive nature of materials – especially metal, which is often used in the vicinity of the optical mode in nanolaser design – needs to be

16

Introduction

considered. Taking these points and the carrier density dependence of the various parameters into account, the rate equations can be written as ∂n I ¼ ηi  Rnr ðnÞ  Rsp ðnÞ  Rst ðnÞS ∂t qVa

ð1:7Þ

∂S S ¼  þ ΓE βðnÞRsp ðnÞ þ ΓE Rst ðnÞS ∂t τp

ð1:8Þ

where n = carrier density (cm−3) I = injection current (A) ηi = current injection efficiency q = electron unit charge (Coulomb) Va = active volume (cm−3) Rnr(n) = non-radiative recombination rate (cm−3∙ s−1) Rsp(n) = total spontaneous emission rate (cm−3∙ s−1) Rst(n) = stimulated emission coefficient (s−1) S = photon density (cm−3) β(n) = spontaneous emission coupling factor τp = photon lifetime (ns) ΓE = energy confinement factor. In different variations of rate equations, the confinement factor Γ has referred to power confinement, electric field confinement, electric energy confinement, and total optical energy confinement [54]. In the context of nanolasers, given the need to account for the negative permittivity and dispersive properties of the metal plasma, the total optical energy confinement ΓE is the only valid expression for confinement factor that can be applied under all circumstances [54]. It is expressed as ð ε0 dr ½εR ðr; ωm Þ þ εg ðr; ωm ÞjEm ðrÞj2 Va 4 V ΓE ¼ ð a ≡ ð1:9Þ ε0 V eff dr ½εR ðr; ωm Þ þ εg ðr; ωm ÞjEm ðrÞj2 4 V εg ðr; ωÞ ¼

∂½ωεR ðr; ωÞ at ω ¼ ωm ∂ω

ð1:10Þ

where εR = real part of the relative permittivity ε εg = real part of the relative group permittivity as defined in Equation (1.10) Va = active region volume (cm−3) Veff = effective mode volume (cm−3) Em(r) = electric field phasor The non-radiative recombination consists of surface recombination that dominates below threshold and Auger recombination that dominates far above threshold. They are

1.5 Laser Rate Equations

(a)

7

12 10 8 6 4 2 0

Figure 1.9

(b)

× 1018

Photon density [cm–3]

Carrier density [cm–3]

14

0

5 Pump power P

× 1016

6 5 4 3 2 1 0

10

17

0

5 Pump power P

× 1028

10

× 1028

Steady-state solution to the rate Equations (1.7) and (1.8) for (a) carrier density and (b) photon density.

Rsurface ðnÞ ¼ vs

Aa n Va

RAuger ðnÞ ¼ Cn3 Rnr ðnÞ ¼ Rsurface ðnÞ þ RAuger ðnÞ

ð1:11Þ

where vs = surface velocity (cm∙ s−1) Aa = surface area of the active material (cm2) C = Auger recombination coefficient (cm6 ∙ s−1) The photon lifetime τp is related to the mode wavelength and Q factor 1 ωk ωk ωk ¼ ¼ þ τp Q Qrad Qabs

ð1:12Þ

where ωk = resonant angular frequency of the k-th mode (rad ∙ s−1) Qrad = quality factor due to radiation Qabs = quality factor due to absorption and scattering loss Figure 1.9 shows a typical steady-state solution to the rate Equations (1.7) and (1.8) in terms of carrier and photon density. The total spontaneous emission rate Rsp(n) consists of spontaneous emission from all discrete cavity modes, in addition to the radiation into the free space continuum of modes. It is expressed as Rsp ðnÞ ¼

X k¼1

Rsp;k ðnÞ þ

X k≠1

ð Rsp;k ðnÞ þ

1 τsp;rad

  dKfc;K 1  fv;K

ð1:13Þ

18

Introduction

where k = 1 denotes the lasing mode, and k ≠ 1 denotes all other cavity modes. The last term in Equation (1.13) accounts for the radiation into the free-space continuum of modes, and it can be modeled as the ratio between the carrier recombination from active medium’s available density of states and the background radiative time τsp,rad [59]. Last, the single mode spontaneous emission rate Rsp,k(n) is related to the total spontaneous emission rate Rsp(n) via Rsp;k ðnÞ ¼ βðnÞ  Rsp ðnÞ

ð1:14Þ

In Equation (1.14), the carrier density dependency of β results from the Purcell effect that is pump dependent: it rises with increasing carrier density until the threshold condition is reached. In the literature, however, the β-factor is usually taken to be independent of the carrier density as we reveal and discuss in detail in Chapter 3. The stimulated emission rate Rst(n) is related to Rsp(n) via Einstein’s A and B coefficients [60]. At steady state, the photon density S is solved from Equations (1.7) and (1.8): S¼

ΓE βðnÞRsp ðnÞ 1=τp  ΓE Rst ðnÞ

ð1:15Þ

Although we have considered this set of rate equations with nanolasers in mind, it is applicable to both metallic and dielectric cavities, from nano- to micro-, and to macroscale lasers. In Equation (1.15), the optical energy confinement factor ΓE, which takes into account the plasma dispersion and the negative permittivity of metal is of crucial importance, because it ensures positive optical energy. Moreover, the Purcell effect in nanolasers is not directly included in the rate equations via the Purcell factor Fp, but rather incorporated via the spontaneous emission factor β that depends on Fp (the derivation of Fp and the relationship between β and Fp is detailed in Chapter 3). β (and Fp)’s influence on the light-light curve and the threshold condition is illustrated in Figure 1.10, which plots the light-light curve for different values of β. Light-light curve is traditionally plotted in linear scale (Figure 1.10(a)), in which a “kink” is visibly seen for the curve with a small β (e.g., β of 0.001 and 0.1). This kink is often referred to as the lasing threshold. However, as β becomes larger, the kink is gradually replaced by an increasingly smoother feature and eventually completely disappears in the extreme case of β = 1. While β is typically on the order of 1×10−5 for conventional semiconductor lasers, the range of values used in Figure 1.10 is typical for nanolasers. To see more clearly the subthreshold behavior, nanolaser light-light curves are usually plotted in log scale, as shown in Figure 1.10(b). Because the kink region is associated with lasing threshold, it is argued that threshold decreases with β (all else being equal), and consequently, a thresholdless behavior is reached when β becomes unity [14] (although this argument has received some controversy). Recently, it has been suggested that lasing threshold can be defined to be when the second-order derivative of the linear lightd d2I light curve reaches a maxima, when ¼ 0 is satisfied; or defined to be when the dI dI 2

1.6 Nanolaser Types and Their Characteristics

(a) 7

(b)

× 1016

1016

5 4 3 β = 0.001 β = 0.1 β = 0.5 β=1

2 1

Figure 1.10

0

2

4 6 Pump power P

8

10 × 1028

Photon density [cm–3]

Photon density [cm–3]

6

0

19

1014

1012

β = 0.001 β = 0.1 β = 0.5 β=1

1010 1025

Pump power P

1030

Simulated light-in vs. light-out curve for different values of spontaneous emission factor β, plotted in (a) linear and (b) log scale.

first-order derivative of the log-scale light-light curve reaches a maxima, when d dlogð LÞ ¼ 0 is satisfied [16]. dI dlogð I Þ

1.6

Nanolaser Types and Their Characteristics Along with device footprint, Figure 1.2 listed several other key parameters for various types of micro- and nanolasers, such as cavity Q factor and lasing threshold. We can see from Figure 1.2 that dielectric and metallic cavities possess distinctively different footprint and Q factors, but they have comparable lasing threshold. In this section, we overview the main breeds of nanolasers, categorizing them either in terms of photonic or plasmonic lasing, or by their cavity geometrical types.

1.6.1

Vertical Cavity Surface-emitting Lasers (VCSELs) Although not a nanolaser, VCSEL is a type of semiconductor micro-scale laser that played major role in the miniaturization of lasers. Proposed in 1979 and first experimentally demonstrated in the 1980s, it took near a decade for VCSEL to evolve from its initial demonstration at cryogenic temperature [61] to CW current injection at room temperature [62]. Along with this breakthrough came many technological improvements, one of which is the monolithic growth of the active material and Bragg mirrors. After this, the development accelerated and VCSELs became commercially available in late 1990s [63], and are now widely used in commercial data communication systems, especially in short-range fiber-optic communications. Figure 1.11(a) shows the schematic of a typical VCSEL structure. VCSEL marks a milestone in laser miniaturization

20

Introduction

(a)

Top DBR tput

Oxide apertures

t ou

Ligh

DBR Topures pert a e d Oxi ty cavi

t ntac p-co a s e m Top

Acti m Botto a s e m

ve a

Active region Bottom DBR ntact con

rea

DBR omte Bott stra Sub

Light output Ag

(b) n-InGaAsP 5.5 pairs of n-DBR

Ag 2 µm

Active region (5 MQWs) 5.5 pairs of p-DBR

SiNx

Au Si Figure 1.11

(a) A typical schematic of a VCSEL structure. The inset shows a scanning electron microscopy image (SEM) of a cross section through the top and bottom DBRs, active region, and multiple oxidation apertures of a 980nm VCSEL. (b) Schematic of the proposed metal-clad VCSEL nanolaser. The thicknesses of the 5.5 pairs of n-doped DBR (n-DBR), the active material, and the 5.5 pairs of p-doped DBR (p-DBR) are 860, 245, and 860 nm, respectively. The whole device is encapsulated in silver, which offers high optical reflectivity and good heat dissipation. Part (b) is reprinted from reference [9] with permission from Optical Society of America (OSA).

by employing high-reflectivity multilayer mirrors − also known as distributed Bragg reflectors (DBRs) – on both sides of the thin MQW active region. Carriers are injected through the metal contacts at the top and bottom of the structure, and light is emitted from the top surface. The current path, from the top to the bottom contact, is defined by insulating, selectively oxidized layers that form an oxide current aperture. The diameter of the oxide aperture primarily governs the beam profile. The high reflectivity (~0.99) of the Bragg mirrors produces high Q cavities with picosecond scale photon lifetime as well as low lasing threshold on the order of micro-amperes. Although the use of DBR

1.6 Nanolaser Types and Their Characteristics

21

provides high reflectivity, the typical tens of layers of Bragg gratings in the DBR design result in a device thickness on the order of tens of micrometers. Furthermore, the lateral dimension is several micrometers for large transverse mode confinement and small surface recombination. Recently, a VCSEL type nanolaser with hybrid mirrors has been proposed (Figure 1.11(b)), in which only 5.5 pairs of n-doped and p-doped DBRs are used in order to reduce the cavity volume. In the meantime, metal cladding is employed to provide moderate optical energy confinement factor while still sustaining a high reflectivity. The total thickness of the structure is 2 μm, which is much less than that of the conventional VCSELs of tens of μm. In addition, the structure is favorable for MQW and QD gain media, since the standing wave enhancement factor can further increase the confinement factor and reduce the threshold gain, which is beneficial for high-speed modulation.

1.6.2

Photonic Crystal Defect Cavity Lasers Photonic crystals are engineered optical nanostructures with refractive indices that vary periodically. First demonstrated in 1998, when designed with a defect, these structures can confine light sometimes to diffraction-limited volumes using 2D or 3D Bragg gratings. Figure 1.12 shows the top view of a defect cavity photonic crystal laser. All photonic crystal lasers are characterized by a photonic bandgap in that light whose wavelengths fall within the photonic bandgap cannot exist in the crystal. The bandgap has similar effect as optical mode selection in a typical Fabry-Perot cavity. Two types of photonic crystal lasers exist, namely, the defect cavity and the broad-area cavity. Figure 1.12 shows a photonic crystal defect nanocavity with a single defect.

Figure 1.12

Schematic of a single-defect photonic crystal nanocavity. Inset: electric field intensity distribution showing that the mode is confined in the defect. Reprinted from reference [64] with permission from John Wiley & Sons, Inc.

22

Introduction

Although often only several hundred nanometers thick, these cavities typically measure tens of micrometers in the transverse direction, because a large number of periods are needed to confine the optical mode in the defect. Therefore, they do not satisfy the nanolaser metric we introduced in Section 1.1. Nonetheless, because of the tightly confined optical mode (inset of Figure 1.12), they find many applications similar to nanolasers with small footprint and small mode volume. These applications include the manipulation of spontaneous emission rate and the study of light-matter interaction. Therefore, they are sometimes categorized as nanolasers [13]. For a summary of some applications enabled by such photonic crystal defect cavity lasers, see reference [64].

1.6.3 1.6.3.1

Nanowire Lasers Photonic Mode Nanowire Lasers Semiconductor nanowire lasers are quasi-1D structures with diameters ranging from tens to hundreds of nanometers and length ranging from a few to hundreds of micrometers. Their width is large enough such that quantum-size effects can be ignored, but optically, they are high-quality 1D waveguides with cross sections in rectangular, triangular, hexagonal, or cylindrical forms. There is usually a high contrast between the refractive indices of the wires (typically 2.5–3.5 of nanowire materials) and the surroundings (typically air with refractive index of 1) in traditional all-dielectric nanowires lasers. Such high contrast makes nanowires good optical waveguides and lasers with moderate energy confinement. Although individual nanowires can work as laser cavities, they do not satisfy the subwavelength requirement in all three dimensions, the nanolaser metric introduced in Section 1.1. Nonetheless, semiconductor nanowire lasers represent a unique development in the history of semiconductor lasers. Interestingly, the major developments in this area during the past decade are largely not from the traditional optoelectronics or semiconductor laser community; the pioneering initial contributions were made by researchers from chemistry and materials communities by developing new nanowires growth and fabrication methods. In 2001, Huang et al. demonstrated the first lasing action in ultraviolet (UV) spectrum from ZnO nanowires (Figure 1.13(a)) via optical pumping with He–Cd laser (325 nm), paving the way to the lasers using 1D nanowires [65]. Since then, optically pumped lasing emission has been observed from a variety of nanowire-like structures of binary semiconductors, including ZnO (∼385 nm), GaN (∼375 nm), ZnS (∼337 nm), CdS (∼490 nm), CdSe (∼710 nm), GaAs (∼850 nm), and GaSb (∼1550 nm) nanowires, and ternary semiconductors including CdSSe and ZnCdS nanowires, with emission wavelength spanning from UV to near IR [66]. Near a decade after the initial demonstration, the field of semiconductor nanowire laser has been revived again as the materials quality improves and more suitable device structures are fabricated, especially using fabrication technologies that are more compatible with the standard III–V device fabrication. Nanowire lasers are conceptually similar to the Fabry-Perot laser that we discussed in Section 1.3, in that the high-index wire acts as the Fabry-Perot cavity and the two ends of

1.6 Nanolaser Types and Their Characteristics

Figure 1.13

23

(a) The first semiconductor nanowires laser demonstrated by Huang et al. with ZnO nanowires. SEM images: (top) zoom-in view and (bottom) zoom-out view. (b) GaAs/AlGaAs/GaAs coreshell nanowires laser. SEM image shows nanowires standing vertically on a tapered base on the growth substrate. Scale bar, 1 mm. Inset: top view of the nanowire, showing the hemispherical Au catalyst nanoparticle atop the hexagonal nanowire. Scale bar, 100 nm. Part (a) is reprinted from reference [65] with permission from American Association for the Advancement of Science (AAAS); part (b) is reprinted from reference [67] with permission from Macmillan Publishers Ltd.

the wire act as end-mirrors of the cavity. A standing wave is formed between the two ends of the wire, and the light leaking out of the wire end becomes the laser output. However, because of the end-facet’s low reflectivity and the poor optical confinement when the wire width is reduced below a few hundred nanometers, and because of the large surface and Auger non-radiative recombination, lasing behavior can only be observed in nanowires many micrometers long under strong optical pumping condition [66]. Novel growth techniques have been investigated to decrease non-radiative recombination but, at the same time, increase mode confinement, for example, by growing core-shell nanowires. Figure 1.13(b) shows a GaAs/AlGaAs/GaAs core-shell nanowire laser that operates at room temperature under optical pumping.

24

Introduction

(b)

(a)

y x

L

z Metal

ksp

ksp-R

(i) γrad = 2/L.InR

Cd

Sn

R

CdS

MgF2

100 nm

Ag

405 nm

an

ow

ire

489 nm

d

kscat

MgF2 CdS

ksp

h

Metal

Metal

(iii) γscatS-I-M interface

log10[Average output power]

Laser

d = 129 nm h = 5 nm 0

1

2

10 10 10 –2 Pump intensity (MW cm ) 1.5 Δλ (nm)

(ii) γscdsurface roughness

SponAmplified taneous Spontaneous emission emission

Metal

Average output power (arbitrary units)

Ag Metal

1.0 0.5 0.0 0.00

0.05

0.10 –1

1/L (μm )

460

470

480

490

500

510

520

530

540

Wavelength (nm)

Figure 1.14

1.6.3.2

(a) Schematics of the main surface plasmon damping mechanisms: radiative emission at cavity facet (γrad, i), scattering at rough metal surface (γscd, ii) and scattering at semiconductor-insulatormetal interface (γscat, iii). (b) Top: schematic and SEM image of the plasmonic laser consists of a CdS semiconductor nanowire on top of a silver substrate, separated by a MgF2 spacer; bottom: its lasing and threshold characteristics. Part (a) is reprinted from reference [70] with permission from Macmillan Publishers Ltd; part (b) is reprinted from reference [26] with permission from Macmillan Publishers Ltd.

Plasmonic Mode Nanowire Lasers Another way to increase the optical mode confinement is through the use of metal. The plasmonic nanowire laser, theoretically proposed and experimentally demonstrated by Oulton et al. [26], consists of a very thin low-index dielectric layer separating a metal surface and a high-index semiconducting nanowire, which functions as the gain medium (Figure 1.14(b)). In this configuration, the low-index region supports a transverse magnetic (TM) polarized photonic and plasmonic mode hybridized in the transverse direction, allowing the propagation of SPPs over long distances [68]. The plasmonic modes resonate between the reflective nanowire end-facets in a Fabry-Perot cavity configuration, similar to the pure photonic mode nanowire lasers discussed in Section 1.6.3.1. There are four principal loss mechanisms: intrinsic ohmic damping, extrinsic radiative emission at the end-facets [69] (γrad, panel (i) in Figure 1.14(a)), extrinsic scattering of propagating SPPs due to surface roughness (γscd, panel (ii) in Figure 1.14(a)), and extrinsic scattering of propagating SPPs at the materials interfaces [27] (γscat, panel (iii) in Figure 1.14(a)). In

1.6 Nanolaser Types and Their Characteristics

25

the first plasmonic nanowires laser demonstration, the strong mode confinement and a high-quality, high-gain material allowed authors in reference [26] to demonstrate deep subwavelength lasing action of SPPs with mode areas as small as λ2/400 (Figure 1.14(b)). However, the length of the laser is still several wavelengths. With improved metal deposition techniques and high-quality crystal growth to reduce the damping losses of the cavity, Lu et al. showed remarkable work developing a smooth epitaxial silver film growth technique together with the growth of InGaN/GaN core-shell hexagonal nanowires with very well-defined facets [27]. The authors demonstrated the first nanowire laser that is subwavelength in all three dimensions, therefore termed a “nanorod” laser. The nanorod has a total length of a mere 480 nm and a width of 50 nm. The device is schematically shown in the top panel of Figure 1.15(a), along with its SEM and scanning transmission electron microscopy (STEM) images showing device dimension and material composition. The cavity is defined by a GaN nanorod (480 nm in length) partially filled with an InGaN gain medium (170 nm in length) and separated by a

Figure 1.15

(a) Top: schematic of a single InGaN-GaN core-shell nanorod on a SiO2-covered epitaxial Ag film of 28 nm thickness, SEM and STEM images of the nanorod indicating the InGaN gain core and GaN shell; bottom: energy distribution of the lasing mode and emission characteristics (light in– light out curve and linewidth evolution) at 8 K and 77 K. (b) Top: all-color InGaN/GaN core-shell nanorod plasmonic laser with SEM image in the inset; bottom: single-mode lasing spectrums and far-field radiation pattern observed from single nanorods from the top panel. Part (a) is reprinted from reference [27] with permission from American Association for the Advancement of Science (AAAS); part (b) is reprinted from reference [71] with permission from American Chemical Society (ACS).

26

Introduction

5-nm silicon dioxide (SiO2) spacer from the epitaxial Ag. The bottom panel of Figure 1.15(a) shows the mode profile as well as the emission characteristics (including light in–light out curve and linewidth evolution) at cryogenic temperatures. This work is a major milestone in reducing nanowire lasers’ size to truly subwavelength. Following up on their own work, the same group of researchers showed InGaN/GaN core-shell nanorod lasing across a broad range of wavelengths in the visible spectrum, by changing the indium (In) content during the InGaN core growth [71]. Figure 1.15(b) depicts the device schematic, SEM, and emission characteristics. In this case, tuning is achieved by varying the gain composition rather than the resonator geometry, thanks to plasmonic mode’s weak dependence on cavity geometry. With most of the work on plasmonic nanolasers discussed so far showed lasing at cryogenic temperatures, Zhang et al. demonstrated in 2014 an efficient room temperature low-threshold plasmonic nanolaser [70]. Although tremendous progress has been made in the field of plasmonic nanowire lasers, electrically pumped versions still remain to be demonstrated. To a large extent, this may be attributed to the difficulty in integrating metal contacts with the nanowire growth for the proper control and injection of charge carriers.

1.6.4

Cavity-free Nanolasers With the two key aspects of a laser being light amplification and feedback, cavities typically play an important role in laser design because they are used to enable feedback. In general, laser cavities are constructed such that photons emitted from the gain medium traverse the cavity many times, giving rise to a large photon cavity lifetime (large cavity Q). However, a cavity is not a prerequisite for enabling feedback. This concept has previously been explored in micro-scale and larger random lasers [72]. In random lasers, feedback arises from multiple random scattering events that lead to the formation of closed optical paths within the disordered medium [73]. A wave making a complete round trip in the closed optical path would return to the starting point with the same phase except for some integral multiple of 2π. Borrowing the design concept from random lasers, cavity-free feedback is possible in plasmonic nanolasers as well. Harnessing the slow/stopped light phenomena to provide local feedback, Pickering et al. designed a cavity-free nanolaser in which photons are trapped and amplified in space at the point of generation, with a theoretically infinite interaction time with the gain medium [74]. Figure 1.16(a) depicts the laser schematic in this theoretical work: it is based on a planar metal-dielectric-metal heterostructures (in the y-direction), with the dielectric layer consisting of gain medium sandwiched laterally between dielectric materials without gain but with a nearly identical real index of refraction as the gain medium. At a particular thickness of the dielectric layer, opposing energy flows in the dielectric layer (with positive permittivity) and in the metal layers (with negative permittivity) are balanced exactly. At the stoplight point, the overall energy flow effectively cancels, forming a closedloop vortex as shown in Figure 1.16(a). In the presence of gain, this stoplight feedback mechanism can lead to coherent amplification of the trapped photons via stimulated

1.6 Nanolaser Types and Their Characteristics

Figure 1.16

27

(a) Subwavelength localized vortex-lasing in a stopped-light design, based on a gain-enhanced plasmonic heterostructure. The stoplight feature of the design allows for the trapping of photons in a closed vortex visualized by paired semicircular arrows. Exploiting leaky-modes, emission into free space takes place perpendicular to the surface. (b) By engineering the band structure of the metal-dielectric waveguide, several stopped-light (SL) points can be made to fall within the gain spectrum, resulting in a flat dispersion with an average slope of (ω2−ω1)/(k2−k1). Reprinted from reference [75] with permission from Royal Society of Chemistry (RSC).

emission. By choosing transparent conductive oxide as the top metal layer, light can be extracted from the top of the cavity. In order to form a stoplight point, the dispersion relation needs to be engineered such that the group velocity is zero at a particular k-point, where k denotes the wavevector. Panel (i) of Figure 1.16(b) shows a stoplight point at k = 0. However, having only one stoplight point is not sufficient in the laser design. By having two stoplight points, a broader range of k-modes falls within the spectrum window of gain such that the mode becomes more localized (Figure 1.16(b), Panel (ii)). Further, to enable subwavelength localization of the lasing mode, it is most desirable for the two stoplight points to align in frequency (Figure 1.16(b), Panel (iii)). In this manner, monotonous behavior of dispersion is enforced between the two stoplight points. The authors of reference [74] exploited this design for operation at near-infrared wavelengths around 1550 nm. The design consists of a top layer of 500 nm thick indium tin oxide (ITO) that simultaneously functions as a metal and window for allowing light extraction, a center layer with a combination of silicon and InGaAsP of 290 nm height, and a bottom layer of thick ITO substrate. For the center layer, silicon is used to minimize absorption of light to the unpumped InGaAsP. Note that although the dielectric

28

Introduction

region is divided into InGaAsP and passive Si sections, no cavity is constructed in the conventional sense because InGaAsP and Si both have a refractive index of 3.4 around 1.55 μm. For now, it suffices to restrict to the passive dispersion relation analysis by taking the entire center dielectric “gain” region to be either lossless InGaAsP or Si, shown schematically in Figure 6(c). Last, a 10 nm Si buffer is added atop the metal. The buffer layer is used to reduce gain quenching in close vicinity to the metal. In theory, the absence of a cavity enables simpler fabrication compared to other plasmonic laser architectures, and the use of the slow-light effect suggests that the threshold gain may be reduced relative to other plasmonic sources. However, because even a few nanometers of interface roughness between the metal and dielectric layer can hugely affect the existence of the stoplight points, the deposition/growth technique for extremely smooth metal surface needs to be developed. Experimental demonstrating of the cavity-free plasmonic laser remains an open problem. We look at mechanisms behind stoplight lasing in more detail in Chapter 9.

1.6.5

Metal-dielectric-metal Waveguide-based Nanolasers The physics behind metal-dielectric-metal (MDM) waveguide and lasers is very rich, and optical modes in MDM-based systems have been investigated in the literature for decades, be it the photonic or plasmonic mode in a three-layer MDM waveguide, or the plasmonic mode that resides at a single metal-dielectric (MD) interface, or the supermode that exists in MD-multilayer with deeply subwavelength thickness. Guided modes in metal-clad waveguide-based nanolasers can be grouped into two main categories: (i) if the real part of the propagation constant, k 0z ðωÞ, satisfies pffiffiffiffiffiffiffiffiffiffiffi k 0z ðωÞ < k0 ðωÞ ε0g ðωÞ, the wavevector projection along the dielectric-metal interface is shorter than that of a plane wave in the dielectric alone, which results purely from reflections within the metal cavity. In this case, the guided modes are photonic modes, which are conventional resonant modes. (ii) If the real part of the propagation constant pffiffiffiffiffiffiffiffiffiffiffi satisfies k 0z ðωÞ > k0 ðωÞ ε0g ðωÞ, the wavevector projection along the dielectric-metal interface is longer than that of a plane wave in the dielectric alone. The guided modes under this condition are surface-bound, plasmonic resonant modes (i.e., surface plasmon polariton (SPP) modes).   SPP modes are highly confined and thus have high Re neff , especially near the SPP resonance where the real part of the wavevector reaches a maximum, as seen in Figure 1.3. The disadvantage of SPP modes is the relatively large mode overlap of the optical field with the metal, which implies high Joule loss, again especially near the SPP resonance where the imaginary part of the wavevector reaches a minimal negative value. One possible solution to the metal loss obstacle is to reduce the temperature of operation. This simultaneously provides two benefits − a reduction of the Joule losses in the metal and an increase in the amount of achievable semiconductor gain. Hill and colleagues [34] demonstrated the first metal-clad semiconductor nanolaser, in which cryogenic lasing was achieved in gold-coated semiconductor cores with diameters as small as 210 nm. Another solution to the loss in metal is to operate the device in a frequency range much below the

1.6 Nanolaser Types and Their Characteristics

29

SPP resonance, where loss is less severe and can be fully compensated by gain [44, 76]. Alternatively, one could operate near the SPP resonance in an MDM structure in which significant modal gain has been shown in certain ranges of material gain values [55]. In the meantime, the negative real parts of the permittivity of metals not only allow them to support SPP modes but also enable them to act as efficient mirrors. This leads to the second class of metal-clad cavity modes that utilizes photonic modes, which can be viewed as lossy versions of the modes in a perfectly conducting metal cavity. Because the overlap between the mode and the metal is usually much smaller for these modes, a cavity supporting this type of modes is able to achieve higher Q factors and lower lasing gain thresholds, albeit at the expense of reduced mode confinement (compared to SPP modes). In general, both types of modes can exist in a metal-clad cavity. We study MDM-based waveguide in more detail in Chapter 2 and MDM-based nanolasers throughout the book. Here, we outline the popular nanolaser geometries based on this design.

1.6.5.1

Metal-clad Nanopillar Lasers The metal-clad nanopillar laser is based on the MDM waveguide. Following the epitaxial growth of semiconductor materials, the pillars are typically defined by standard dry-etching techniques. Subsequent dielectric and/or metal layers are then introduced by thin-film deposition and/or sputtering. In terms of the supported optical modes, in the MDM waveguide direction, either photonic or plasmonic mode can be supported depending on the diameter of the dielectric layer. Perpendicular to the waveguide direction, a Fabry-Perot–like cavity is formed and light is emitted from its partially reflective end-facet. Because of the ease in fabrication to define the MDM waveguide, and the versatility in forming the FabryPerot cavity perpendicular to the waveguiding direction, metal-clad nanopillar lasers have become one of the most popular geometry in semiconductor nanolaser design. For example, an optically pumped laser would have a Fabry-Perot cavity consisting of metal-gain-metal. By changing the Fabry-Perot design to metal-semiconductor-gain-semiconductor-metal, an electrically pumped version is conceived. Indeed, this is one of the few types of nanolasers in which electrically pumped designs are exploited, and CW electrically pumped operation at room temperature is demonstrated [37]. In terms of footprint, the height of the pillars can be easily controlled to be within the emission’s free-space wavelength even in the case of current injection; the width of the pillars is diffraction limited to λ0 =2neff , where neff ≈ ngain if the supported mode is photonic in nature and neff > ngain if the supported mode is plasmonic in nature. In terms of optical mode volume, they are always less than the device footprint, thanks to metal’s ability to tightly confine the mode. In 2007, Hill et al. experimentally demonstrated the first nanolaser of this type, shown in Figure 1.17(a) [34]. Being electrically pumped and operating at cryogenic temperature, it remains to be one of the smallest devices of this kind. Since then, various efforts have been made to increase operating temperature and efficiency and decrease threshold gain as well as device size. Figure 1.17 summarizes a few signature experimentally demonstrated nanopillar lasers, with (a−c) being electrically pumped and (d) being optically pumped. These nanolasers are studied in more detail in later chapters.

30

Introduction

Figure 1.17

(a) Electrically pumped cylindrical nanolaser by Hill et al. [34]. Top: laser schematic, bottom: SEM image of device before metal-cladding formation. (b) Electrically pumped rectangular nanolaser by Ding et al. [37]. Top: laser schematic, center: SEM image of device after dry etching, bottom: SEM image of completed device. (c) Electrically pumped cylindrical nanolaser with undercut by Gu et al. [38]. Top: laser schematic, bottom: SEM image of device after undercut of doped regions. (d) Optically pumped cylindrical nanolaser by Nezhad et al. [35]. Top: laser schematic including the optical pumping and laser emission directions; bottom: SEM image of device before metal-cladding formation. Part (a) reprinted from reference [77] with permission from John Wiley & Sons, Inc. and reference [34] with permission from Macmillan Publishers Ltd; part (b) reprinted from reference [37] with permission from Optical Society of America (OSA); part (c) reprinted from reference [38] with permission from Institute of Electrical and Electronics Engineers (IEEE); part (d) reprinted from reference [35] with permission from Macmillan Publishers Ltd.

1.6.5.2

Metal-clad Antenna-inspired Nanolasers In addition to its usefulness in confining optical mode and reducing device footprint, metal can also be used to control the radiation pattern of emitted light in nanolasers. In the context of chip-scale integration, while unwanted cross talk should always be avoided, it is also desirable to efficiently channel useful information between components. For example, how can the emitted light from the on-chip laser be efficiently coupled to a waveguiding component? To this end, techniques used in the microwave community are borrowed and the antenna-inspired nanolaser is conceived. The traditional understanding of antennas comes from their radio frequency (RF) developments and their applications in our everyday life such as those to transmit or receive radio signals. In a nutshell, an antenna acts as an efficient transducer to convert unbound electromagnetic waves propagating in free space into confined electromagnetic fields with desired radiation patterns. Comparing RF and optical frequency antennas, one of the major differences comes from the metal property and response in the two

1.6 Nanolaser Types and Their Characteristics

31

frequency ranges. With metals being good conductors characterized by the deeply subwavelength skin depth at RF frequency, it is not the case at optical frequency. This is particularly so as the plasma frequency of metals is approached. Nonetheless, the field of optical antenna has rapidly grown during the past decade, with the aim in applications such as bio-sensing, optical radiation pattern control, energy harvesting, and imaging [78]. For semiconductor nanolasers, the nanopatch antenna laser (a few of its configurations are schematically shown in Figure 1.18) has received particular attention. Optically pumped nanopatch of the cylindrical type has been experimentally demonstrated (Figure 1.18(d)) [79], and the electrically pumped version has been designed [80, 81] but has yet to be demonstrated. If not restricted to semiconductor lasers, various types of nano-antenna laser geometries have been investigated, a few example geometries of which are shown in Figure 1.19. The most studied type is the bowtie geometry (Figure 1.19(b) and (c)), and Suh et al. experimentally showed plasmonic lasing behavior in 3D bowtie laser arrays with organic dye as the gain medium; 2D bowtie lasers have also been investigated in III-V semiconductor systems, although the device height has been capped at several micrometers and only electro-luminescence has been observed [83].

1.6.5.3

Other Metal-clad Nanolasers Apart from the mainstream semiconductor nanolasers discussed earlier, various other types have been investigated and some experimentally demonstrated. Kwon et al. constructed a nano-pan laser (Figure 1.20(a)), which utilizes a whispering-gallery cavity at the semiconductor-metal interface in a silver-cladded InP/InAsP/InP disk of 235 nm thick [84].

(b)

(c)

Metal

Gold

Metal p gain n

p gain n

(d)

Metal p gain n

220 nm

(a)

InGaAsP

Substrate Figure 1.18

Substrate

200 nm

430 nm

Examples of nanopatch laser structures. (a) Circular nanopatch laser. (b) Rectangular nanopatch laser. (c) Hexagonal nanopatch laser. The structures are not drawn to scale. (d) Optically pumped nanopatch lasing that operates at 77 K. Parts (a)−(c) reprinted from reference [80] with permission from Institute of Electrical and Electronics Engineers (IEEE); part (d) reprinted from reference [82] with permission from Institute of Physics (IOP) Publishing.

(a)

Nanorod dimer Figure 1.19

Substrate

(b)

(c)

2D bowtie

3D bowtie

Examples of nano-antenna laser geometries: (a) nanorod dimer, (b) 2D bowtie, and (c) 3D bowtie.

32

Introduction

Figure 1.20

(a) Schematic of the plasmonic nanopan cavity and SEM images of the nano-pan laser showing the InP disk on glass prior to silver deposition and the silver film separated from the disk [84]. (b) Schematic of a square plasmon laser showing a thin CdS square atop a silver substrate separated by a 5 nm MgF2 gap, where the most intense electric fields of the device reside and an SEM image of the CdS square plasmon laser that is 45 nm thick, 1 μm in length [85]. (c) Schematic of a nano-coaxial laser cavity and SEM images of the constituent rings. The side view of the rings comprising the coaxial structures is seen. The rings consist of SiO2 on top and a quantum well gain region underneath [14]. Part (a) reprinted from reference [84] with permission from American Chemical Society (ACS); part (b) reprinted from reference [85] with permission from Macmillan Publishers Ltd; part (c) reprinted from reference [14] with permission from Macmillan Publishers Ltd.

This cavity exhibited lasing behavior up to 80 K under optical pumping. Ma et al. reported a nano-square laser (Figure 1.20(b)) operating at room temperature under optical pumping [85]. The device consisted of a 45 nm thick, 1 μm long single crystal cadmium sulfide (CdS) square atop a silver surface separated by a 5 nm thick magnesium fluoride (MgF2) gap layer. Strong feedback is achieved via total internal reflection of surface plasmons at the cavity boundaries, resulting in sufficiently high Q-factors. The utilization of this feedback mechanism also filters out all photonic modes as they have insufficient momentum for total internal reflection, leaving only the plasmonic ones as viable cavity mode candidates. The combined efforts of strong confinement, low metal and radiation loss therefore led to the room temperature operation in this metal-insulator-semiconductor (AgMgF2-CdS) nano-square laser. Last but not least, the nano-coaxial laser (Figure 1.20(c)) by Khajavikhan et al. represents a MDM waveguide – Ag/InGaAsP/Ag with InGaAsP width of 100 nm and waveguide length of 1.25 μm – but wrapped around with the two end-facets of the waveguide connected [14]. The cavity resembles the conventional coaxial cable that supports the cut-off-free TEM mode, and the mode energy is concentrated at the metaldielectric interface. This coaxial nanolaser showed lasing action at 4.5 K under optical pumping and featured unity-spontaneous emission factor, sparkling a number of studies on “thresholdless” lasing and laser threshold analysis [16, 86]. We note that a number of nanolasers discussed in previous sections exhibit hybrid photonic and plasmonic mode lasing, such as the nanowires laser in Section 1.6.3.2 (Figure 1.14), while some exhibit pure surface plasmonic mode lasing, such as the

1.6 Nanolaser Types and Their Characteristics

33

nano-square laser in this section (Figure 1.20(c)). This shows the trend from photonic to plasmonic lasing as the laser size becomes smaller, and as a result, the propagation constant of supported modes become larger, as shown in Figure 1.3. In the extreme case where the propagation constant approaches its maxima, pure plasmonicity is reached and SPASER is proposed to explain the physical mechanism that governs emission under this circumstance [46], which is the topic of our discussion in the following section.

1.6.6

SPASERs In 2003, Bergman and Stockman proposed a mechanism to generate temporally coherent surface plasmon modes strongly localized in a V-shaped metallic inclusion plane surrounded by a gain media, shown in Figure 1.21(a) [46]. This mechanism

(a) En (V/m)

5.E7

30 x

z 30

NQDs

e–h pairs

Dielectric core Exciton Silver shell

Energy transfer

(b)

Plasmon

10–20 nm NQD Figure 1.21

Nanoshell

(a) Local field amplitude of coherent surface plasmons in a V-shaped metallic inclusion plane where the dimensions are in nanometers. (b) Left: Schematic of a SPASER made from a metallic nanoshell on a dielectric core, and surrounded by nanocrystal quantum dots; right: schematic of the respective energy levels and transitions. Part (a) reprinted from reference [46] with permission from American Physical Society (APS); part (b) reprinted from reference [87] with permission from Macmillan Publishers Ltd.

34

Introduction

(a)

(b) OG–488 dye doped silica shell

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570 610 Wavelength (nm)

650

(a) Top: schematic of the hybrid nanoparticle architecture (not to scale), indicating dye molecules throughout the silica shell; bottom: stimulated emission spectra of a nanoparticle sample for five pumping powers, in which left inset is the SEM image of Au/silica/dye core-shell nanoparticles and right inset is the spaser mode for a single nanoparticle. (b) Top: plasmonic waveguide (Au-film) sandwiched between optically pumped InGaAs quantum-well gain media. The stacks are immersed in the evanescent field of the long-range SPP mode represented by the transverse optical magnetic field hx. Arrows to the right indicate the diagnostic radiation emitted when SPPs reflect from the end facet; bottom: end-on view showing the ridge architecture after dies are flip-chip bonded. Part (a) reprinted from reference [88] with permission from Macmillan Publishers Ltd; part (b) reprinted from reference [90] with permission from Optical Society of America (OSA).

allows surface plasmon amplification by stimulated emission of radiation, and such devices are termed SPASER. The surface plasmon field close to the vicinity of a metal oscillates and induces stimulated transitions in the gain media close to the metal. In turn, the gain media amplify the localized surface plasmon field, leading to their stimulated emission. Figure 1.21(b) illustrates the SPASER mechanism that in principle is similar to a laser: there is an amplifying medium within a resonant cavity to provide feedback. The main difference between SPASER and laser is that plasmon modes are considered dark modes without out-coupling to the far-field. However, if the dielectric cavity allows, the plasmon oscillations can couple into the photonic modes emitting to the far field [87]. Noginov et al. showed the first experimental demonstration of a SPASER [88]. The authors used nanoparticles consisting of a gold core surrounded by a silica shell

1.6 Nanolaser Types and Their Characteristics

35

containing an organic dye and presenting a dipole-like mode. The spectral evolution, the nanoparticles, and the mode profile are shown in Figure 1.22(a). In this case the feedback mechanism is provided by the surface plasmon oscillating modes. Later, Zheludev et al. proposed the lasing SPASER, a two-dimensional array of asymmetric split-ring resonators, with in-phase collective oscillations of antisymmetric currents, supporting coherent current excitations with high-quality factor with capability of emission into the free space [89]. On an integrated semiconductor platform, the first room temperature spasing was demonstrated by Flynn et al. in 2011, by sandwiching a gold-film plasmonic waveguide between optically pumped InGaAs quantum-well gain media, as shown in Figure 1.22(b) [90]. Experimental demonstration of spasing was also realized by Suh et al. in 2012 with a room temperature nanolaser based on 3D Au bowtie supported by an organic gain material [91]. In parallel to works highlighted here, many others have aimed to understand the linewidth enhancement of SPASERs [92], reach lower power thresholds [93], obtain tunable surface plasmons resonances [94], and use new materials such as graphene [95]. Among all the SPASERs discussed, only the work by Flynn et al. [90] uses semiconductor gain medium. Indeed, most SPASERs to date use organic dye as the active medium because of its higher gain, easier integration with metal, and easier fabrication processes. While SPASER emission shows pure plasmonic behavior (Figure 1.3), lasers in which plasmonic mode dominates the lasing behavior are sometimes also loosely categorized as SPASERs [43]. By this token, the plasmonic semiconductor nanowires lasers in Section 1.6.3.2 and the metal-clad semiconductor nano-antenna and nanosquare lasers in Section 1.6.5.2 are SPASERs.

2

Photonic Mode Metal-dielectricmetal–based Nanolasers

In this chapter, we consider nanolaser design, starting from commonly used optical cavity design and material compositions of conventional dielectric lasers. After the demonstration of whispering gallery mode (WGM) dielectric disk microlasers [18], an intuitive method to further reduce device size is to reduce the azimuthal mode number of WGM disks. Indeed, it has been demonstrated that the diameter of thick (λ0/n) microdisk lasers can be reduced below their free-space emission wavelength [21]. Owing to the small disk diameters, however, the spatial spread of the resultant low azimuthal number modes into the surrounding space is way beyond the physical boundaries of the disks. As a result, it may lead to mode coupling and formation of “photonic molecules” in closely spaced disks [96]. For illustration purposes, Figure 2.1(a) shows a semiconductor disk with radius rc = 460 nm and height hc = 480 nm, and an M = 4 WGM is shown in Figure 2.1(b), clearly indicating the radiative nature of the mode and its spatial spread, which, as mentioned, can lead to mode coupling with nearby structures. As we discussed in Chapter 1, one approach for alleviating the issue of undesired mode coupling is to incorporate metals into dielectric cavity structures, because metals can suppress leaky optical modes and effectively isolate them from their neighboring devices. Embedding the aforementioned gain disk in a gold shield (Figure 2.1(c)) effectively confines the resonant modes at the cost of increasing Joule losses. For approximately the same free-space wavelength, the surface plasmonic polariton (SPP) mode (Figure 2.1(e)) has both a higher M number and higher losses (M = 6, Q = 36) compared to the photonic mode (Figure 2.1(d), M = 3, Q = 183). It should be noted that even though the metal cladding is the source of Joule loss, the large refractive index of the semiconductor core (nsemi ≈ 3.4) aggravates the problem and increases both the plasmonic and Fresnel reflection losses. For SPP propagation on a (planar) semiconductor-metal interface, the threshold gain for lossless propagation is proportional to n3semi [44]. As such, even though SPP modes with relatively high Q can exist inside metal cavities with low-index cores (e.g., silica, with n = 1.48), using this approach to create a purely surface bound plasmonic, room-temperature semiconductor laser at telecommunication wavelengths becomes challenging, due to the order-of-magnitude increase in gain threshold.

2.1

Metallo-dielectric Cavity Design One possible solution for overcoming the metal loss obstacle is to reduce the temperature of operation. This will have two coinciding benefits, a reduction of the Joule losses

2.1 Metallo-dielectric Cavity Design

Figure 2.1

37

The M = 4 whispering gallery resonance for a thick semiconductor disk (a) is shown in (b): (rc = 460 nm, hc = 480 nm, nsemi = 3.4). Note the spatial spread of the mode compared to the actual disk size. (c) The same disk encased in an optically thick (dm = 100 nm) gold shield will have well-confined reflective (d) and SPP (e) modes but with much higher mode losses. |E| is shown in all cases and the section plane is horizontal and through the middle of the cylinder. Reprinted from reference [35] with permission from Macmillan Publishers Ltd.

in the metal and an increase in the amount of achievable semiconductor gain. Such benefits were indeed used in the metal-clad nanolaser demonstrated by Hill et al. [34], which operated at a temperature of 77 K. In this work, metal was directly deposited around the semiconductor core (with a 10 nm SiN electrical insulation layer in between). As a result of the large overlap of the mode with the metal, the estimated roomtemperature cavity Q is quite low. Using a silver coating and ignoring non-radiative recombination, the cavity Q is 180, which corresponds to an overall gain threshold of 1700 cm−1. The gain coefficient reported for optically pumped bulk InGaAsP emitting at 1.55 μm is reported to be 200 cm−1 [97], while InGaAsP multiple-quantum-wells (MQWs) can have gain coefficients more than 2000 cm−1 at high pump levels [17]. It is then fair to say that it would be challenging to achieve room-temperature lasing with the same approach and a similarly sized cavity, due to the constraints imposed by the amount of available semiconductor gain and metal losses. Furthermore, even if the required gain is achievable at room temperature, efficient operation of the device would still be a challenge due in part to thermal heating and in part to non-radiative recombination processes (e.g., Auger recombination). In particular, if a densely packed array of such devices was to be operated (e.g., for applications in chip-scale optical communication), thermal management would be a major concern, given the requisite intense pumping levels. Consequently, it is extremely important to

38

Photonic Mode Metal-dielectric-metal–based Nanolasers

(a)

(b)

(c)

Plug region

εm Rout

εg

εs εg Rg εm

εg

Gain region

εm

εs εs

z r

Figure 2.2

(a) Cross section of the metal-coated composite gain waveguide. (b) Cylindrical closed 3D resonator. (c) Cylindrical open 3D resonator. Reprinted from reference [99] with permission from Optical Society of America (OSA).

optimize the resonator design such that the gain threshold is minimized. In the reminder of this section, we look at approaches to reduce losses in metal-dielectric-metal (MDM) waveguides and cavities. Consider a particular type of MDM waveguide − a composite gain waveguide (CGW) consisting of a gain medium cylindrical core, a dielectric “shield” layer, and a metallic cladding, as shown in Figure 2.2(a). For a given CGW cross-section size, the shield layer thickness can be tuned to increase the confinement of the electric field in the gain medium and reduce the field penetration into the metal. In so doing, one can increase the ability of the device to compensate for the dissipated power with power generated in the gain medium. A direct measure of that is the threshold gain, that is, the gain required for lossless propagation in the CGW. The field attenuation in the shield layer resembles that of Bragg fibers [98]. The layer adjacent to the core, in particular, is of high importance and is also used to reduce loss in infrared hollow metallic waveguides. The CGW model can subsequently be used for the design of subwavelength 3D resonators. To confine the light in the longitudinal direction, the CGW is terminated from both sides by a low index “plug” region covered with metal, which forms the closed cylindrical structure shown in Figure 2.2(b). A more practical nanolaser configuration from a fabrication point of view is the open structure with a SiO2 substrate shown in Figure 2.2(c). The inherent radiation losses into the substrate provide means for collecting the laser light, in contrast to the closed structure, where extracting light requires modification of the metal coating, such as making an aperture in it. The threshold gain for the 3D resonator is defined as the gain required to compensate for the metal losses in the closed structure (Figure 2.2(b)), or to compensate for both the metal and the radiation losses in the open structure (Figure 2.2(c)). In an open structure, the condition under which the threshold gain is defined, namely, the lossless propagation condition, is also known as the transparency condition.

2.1 Metallo-dielectric Cavity Design

39

Let us first consider the infinite CGW of Figure 2.2(a), with relative permittivities εg ¼ ε0g þ jεg00, εs and εm ¼ ε0m  jεm00 of the gain medium, the shield layer, and the metal, respectively. Assuming a time dependence of expð jωtÞ, we have εg00; εm00 > 0. The radius of the gain medium is Rg, the shield layer thickness is Δ = Rout − Rg, and the metallic coating layer begins at radius Rout. The eigenmodes of the CGW may be derived from the general solution of the longitudinal fields in each layer having the form U ¼ ½AJm ðkr rÞ þ BYm ðkr rÞ f ðmφÞejβz , where U = Ez or Hz; Jm and Ym are Bessel qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi functions of the first and second kind, respectively; kr ¼ k02 ε  kz2 , k0 = ω/c; ε is the relative permittivity of the layer; and f ðmφÞ may be expanded by expð jmφÞ, where the integer m is the azimuthal index. The dispersion relation can be found using the transfer matrix method [98]. For εg00 corresponding to the threshold condition, the propagation constant kz is real, and the threshold gain explicitly satisfies ÐÐ ÐÐ εg00 ¼ εm00 Metal dAjEj2 = Gain dAjEj2 , where the integration in the numerator and denominator is over the cross section of the metal, and each propagation mode may be found by imposing Imfkz g ¼ 0 in the dispersion relation and then finding the solutions in the    plane of Re kz g; εg00 , similar to that done in reference [100]. Assuming a wavelength of λ = 1550 nm, ε0g ¼ 12:5 corresponding to InGaAsP gain medium, ε0s ¼ 2:1 for a SiO2 shield layer, and ε0m ¼ 95:9  j11:0 for a gold coating [101], the effect of the shield layer on the TE01 mode threshold gain can be calculated. The result is shown in Figure 2.3(a), where εg00 is plotted as a function of the shield thickness Δ for a given radius Rout = 460 nm. The rapid field decay in the gold layer permits us to assume that the metal extends to infinity, whereas in reality a coating layer of 100 nm would suffice. As the shield thickness increases, a lower percentage of the field penetrates into the metal, reducing the losses. On the other hand, the gain material occupies less of the CGW volume, which means that a higher gain is required to compensate for the dissipation losses in the metal. The trade-off between these two processes results in an optimal point in which the threshold gain is minimal. This

(b)

0.02

Threshold gain (ε″g)

Threshold gain (ε″g)

(a)

0.015

0.01

0.005

0.07

0.06

0.05

0.04

Rout = 460 nm 0

0

50

100

Rout = 300 nm 150

Δ [nm] Figure 2.3

200

250

0.03

0

50 Δ [nm]

100

(a) Threshold gain εg00 as a function of the shield thickness Δ for the TE01 mode with Rout = 460 nm. (b) Threshold gain εg00 as a function of the shield thickness Δ for the TE01 mode with Rout = 300 nm. Reprinted from reference [99] with permission from Optical Society of America (OSA).

40

Photonic Mode Metal-dielectric-metal–based Nanolasers

100 TM01 10–1

TE01 HE11

Minimal ε ′′g

10–2

HE21

10–3 10–4 10–5 10–6 250

Figure 2.4

HE11 plug cutoff 300

TM11 plug cutoff 350

400

HE21 plug cutoff

450 500 Rout [nm]

TE01 plug cutoff 550

600

650

Minimal threshold gain as a function of Rout. The vertical lines show the cutoff of each mode in the 3D resonator plug region. Reprinted from reference [99] with permission from Optical Society of America (OSA).

typical behavior of low-order modes is seen in Figure 2.3(a) for the TE01 mode. For Rout = 460 nm, the improvement of the threshold gain from the Δ = 0 (no shield layer) case is by a factor of 6.1. For larger radii, a lower threshold gain may be achieved. In Figure 2.4, the minimal threshold gain εg00 is depicted as a function of Rout for four low-order modes: TM01, TE01, HE11, and HE21. Having the highest confinement around the gain medium core, the HE11 has the lowest threshold gain among the four modes. Generally, for small radii, the shield layer is less effective as it quickly gets the mode below cutoff. For large radii, the threshold gain is low, as the field penetration into the metal is small. The optimal shield layer thickness increases monotonically as a function of Rout. For the TE01 mode, it ranges between 80 and 330 nm, for Rout = 250 to 650 nm, respectively. The role of the metal coating, which is important in the infinite CGW model, becomes even more crucial for creating a resonant cavity where the mode is confined in all three spatial dimensions. As explained earlier, the CGW facets are terminated by the “plug” regions, which are essentially short metallic waveguides filled with SiO2, as seen in Figure 2.2(b) and Figure 2.2(c). The plug ensures strong confinement of the field in the gain region, provided that the mode residing in it is below cutoff, that is, decaying exponentially in the z direction. For the plug region waveguide, the cutoff is not clearly defined since the modes are significantly different from those of the perfect conducting cylinder waveguide. A reasonable definition for the cutoff is a waveguide with the radius Rout supporting a mode whose kz is closest to the origin on the complex kz plane. That cutoff is shown for each one of the modes in Figure 2.4 by the vertical lines, providing a qualitative tool for choosing an operation mode for the entire 3D structure, as the chosen radius needs to be

2.1 Metallo-dielectric Cavity Design

41

to the left of the vertical line corresponding to the operation mode. The smaller the device radius compared to the cutoff radius, the stronger the decay in the plug; consequently, the threshold gain is lower. While the HE11 mode achieves the lowest threshold gain for a given Rout, its cutoff in the plug region is at a small radius; working below this cutoff entails a relatively high threshold gain. It is therefore seen that the TE01 mode, which has the highest cutoff of the shown modes, is favorable. The result shows that modes corresponding to a larger Rout will have a significantly lower threshold gain. Another advantage of the TE01 is that this mode in the gain region couples only to symmetric TE modes in the plug region, whereas m > 0 modes are hybrid and may couple to all modes with the same azimuthal index. Using the CGW model at the optimal point of Figure 2.3(a) as a starting point, a 3D closed resonator with Rout = 460 and 100-nm-thick gold coating can be designed for the TE012 mode. Using numerical simulation tools, for example, a 3D Finite Element Method (FEM) eigenfrequency solver, the electric field intensity jEj2 normalized to its maximal value can be obtained, as shown in Figure 2.5. The overall height of the resonator is 1500 nm, and the overall diameter is 1120 nm, making it smaller than the vacuum wavelength in all three dimensions. The resonance can be fine-tuned to wavelengths around 1550 nm by setting the gain cylinder height to be about 480 nm and the shield layer thickness to about 200 nm, which is close to the 190 nm predicted by the CGW model. The threshold gain, however, is in less agreement with the CGW model; the value for the 3D resonator is εg00 ffi 0:011, which corresponds to about 130 cm−1, whereas the CGW model gives about 36 cm−1. This discrepancy is due to the losses occurring in the plug region and the mode deformation at the interfaces between the plug

Figure 2.5

Cross section of a closed cylindrical 3D subwavelength laser resonator. The electric field intensity jEj2 normalized to its maximal value of the TE012 mode is shown. The inset shows a similar open structure. Reprinted from reference [99] with permission from Optical Society of America (OSA).

42

Photonic Mode Metal-dielectric-metal–based Nanolasers

and gain regions, two effects that are not taken into account in the CGW model. It is evident that the longer the resonator, the more accurately the CGW model describes the behavior in the gain region. For instance, a longer resonator with the same radius designed for the TE013 mode has a threshold gain of about 95 cm−1. If the structure shown in Figure 2.5 is designed with no shield layer in the gain region, but with the same overall radius and height, then the resulting threshold gain is about 420 cm−1. Given that the gain that may be achieved at room temperature by optical pumping of bulk InGaAsP is about 200 cm−1, it is therefore evident that the shield layer that lowers the threshold gain from 420 to 130 cm−1 is crucial to enable lasing at room temperature for a cavity of this size. Slightly modifying the structure for the open configuration, as shown in the inset of Figure 2.5, the field distribution remains nearly unchanged, and the threshold gain increases only to about 145 cm−1 owing to the radiation losses. The quality factor of this open resonator at transparency, corresponding to the gain threshold of 145 cm−1, is 1125. Before we conclude the conceptual design section, note that we have expressed the threshold condition in terms of the imaginary part of the permittivity, εg00. Alternatively, 00 are the threshold condition can be expressed in terms of gain threshold, gth ⋅ gth and εg;th related by gth ¼

00 2πεg;th λ0 neff

ð2:1Þ

where neff is the effective refractive index of the optical mode. Sometimes, εg00 is more commonly used in the physics community, while gth is more commonly used in the engineering community.

2.2

Invariance of Optimal Metallo-dielectric Waveguide Geometry with Respect to Metal-cladding Permittivity In Section 2.1, gold is used as the metal-cladding in designing the CGW. The choice of metal-cladding affects laser design, fabrication, and performance in several ways. Perhaps most obviously, different metals exhibit varying degrees of loss or, equivalently, the imaginary part of the electric permittivity, εM00 , differs for each metal. Additionally, metals adhere differently to the shield layer − usually SiO2 or SiN [35, 37], react differently with etchants, and exhibit differing stability to the ambient environment. Thus, the ability to predict the behavior of a given CGW structure for a wide range of possible metal-claddings holds significant value. Consider the metallo-dielectric design for a cylindrical CGW, its material and geometrical properties, as well as refractive index and electric field profiles, as described by Figure 2.6. The semiconductor core, lossless dielectric shield, and metal-cladding are characterized electrically with permittivities εG = εG 0 + jεG00 , εD, and εM = εM 0 – jεM00 , respectively. The width of the shield layer ΔD is given by the difference ΔD = Rtotal – Rcore. In general, the electromagnetic field inside such a CGW may be expressed as an infinite, discrete

2.2 Invariance of Optimal Metallo-dielectric Waveguide Geometry

Figure 2.6

43

Re(sqrt(εR)) and |E| of the TE01 mode as functions of radial distance in an optimized CGW at λ0 = 1.55 μm. The permittivities are εG = 11.56 + j8.65 ×10−4, εD = 2.16, and εM = −130 − j3.0, respectively. Reprinted from reference [102] with permission from Optical Society of America (OSA).

sum of solutions to the source-free wave equation, the natural modes of the CGW. Assuming the CGW consists of nonmagnetic materials, each mode may be described by its complex wave-number k, defined as k2 = εRk02 = εR∙ω2/c2 = εR(2π/λ0)2 = βρ2 + βz2 where, generally, the relative permittivity, εR, and the transverse and longitudinal propagation constants, βρ and βz, respectively, are all complex, and λ0 is the free-space wavelength. Along with εR, βρ differs within each layer of the CGW, whereas βz remains constant everywhere for a given mode. The boundary-value problem to be solved consists of finding the eigenvalues to the system of transcendental equations that describes the CGW. Here, we work under the CGW’s threshold condition, βz00 = 0, and the eigenvalues correspond to the zeros in the complex (εG00 , βz 0 ) plane. We are concerned with the TE01 mode because this mode exhibits more favorable properties for use in a nanolaser than neighboring modes, as analyzed in Section 2.1. The threshold gain εGth00 is the value of εG00 necessary to offset the metal loss and make the imaginary part of the propagation constant vanish, that is, εGth00 = εG00 (βz00 = 0). It is related to the material threshold gain per unit length, gth, gth ¼ 2π

00 εgth λ0 ng

ð2:2Þ

where ng is the group refractive index [103]. We can see that, all else equal, a more lossy metal will lead to a larger threshold gain. However, what is not immediately clear is the effect of the metal loss on the optimal shield width. Intuitively it seems that, all else equal, a CGW with a high-loss metal, such as aluminum at room temperature, |εM00 | >> 1,

44

Photonic Mode Metal-dielectric-metal–based Nanolasers

Figure 2.7

Threshold gain εGth00 as function of Rcore for two values of εG 0 , with εM parameterized. Rectangles and circles indicate (Rcore, εGth00 ) = (Rcore,opt, εGth,opt00 ) for InGaAsP and GaS CWGs, respectively. εD, λ0, and Rtotal are fixed at 2.16, 1.55μm, and 0.45μm, respectively. Reprinted from reference [102] with permission from Optical Society of America (OSA).

necessitates a thicker optimal shield for the TE01 mode than a low-loss metal, such as silver, or aluminum at a lower temperature, |εM00 | ~ 1. However, such intuition is wrong. The optimal shield width increases discontinuously, from zero when εM00 = 0, to a constant, non-obvious value for all εM00 > 0. With a fixed Rtotal, and varying only εM, we observe that the optimal shield width ΔD,opt, or equivalently the optimal core radius Rcore,opt, is constant with respect to changes in εM00 and nearly constant with εM 0 . These results are shown in Figure 2.7, where εGth00 is plotted as a function of Rcore with εM and εG 0 as parameters. Constants include εD = 2.16, λ0 = 1.55 μm, and Rtotal = 0.45 μm. The chosen value of Rtotal is sufficiently large to yield relatively low εGth00 but also sufficiently small to yield a relatively high spontaneous emission factor for a laser cavity based on this CGW. The parameterized metal permittivities are –130 – j3.0 (bold line), –130 – j0.3 (dash line), and –260 – j0.3 (dash-dot line), approximately representative of silver at room temperature, silver at liquid nitrogen temperature, and aluminum at liquid helium temperature, all near λ0 = 1.55 μm [69, 104, 105]. The two values of εG 0 are 11.56 (rectangles) and 6.76 (circles), representative of InGaAsP and GaS, respectively [106]. With the order of magnitude reduction in εM00 , Rcore,opt of the InGaAsP (GaS) CGW remains constant at 0.272 μm (0.313 μm) and changes by less than 1% (2%), with the doubling of |εM 0 |. Equivalently, ΔD,opt remains constant with εM00 and varies from 0.178 μm to 0.180 μm (0.137 μm to 0.141 μm) with |εM 0 |. Consistent with the reasoning that a less lossy metal requires less compensation from the gain medium, we further observe that an order of magnitude change in εM00 causes an order of magnitude

2.2 Invariance of Optimal Metallo-dielectric Waveguide Geometry

Figure 2.8

45

neff 0 as function of Rcore for two values of εG 0 , with εM parameterized. Rectangles and circles indicate (Rcore, neff) = (Rcore,opt, neff,opt) for InGaAsP and GaS CWGs, respectively. εD, λ0, and Rtotal are fixed at 2.16, 1.55 μm, and 0.45 μm, respectively. Reprinted from reference [102] with permission from Optical Society of America (OSA).

reduction in εGth00 for both CGWs. Finally, we see that as |εM 0 | is increased by a factor of two, εGth00 decreases by a factor of 2.53 (2.63). Accompanying the invariance of Rcore,opt with respect to εM00 and its weak dependence on εM 0 , the real part of the optimal effective index, neff,opt 0 , where neff = βz(2π/λ0) and neff,opt = neff (Rcore,opt), similarly exhibits invariance and weak dependence upon εM00 and εM 0 , respectively. Figure 2.8 shows that neff,opt 0 remains constant as εM00 is reduced by an order of magnitude. When |εM 0 | is doubled, neff,opt 0 changes by less than 1% (2%), for the InGaAsP (GaS) CGWs. When considering a larger range of εG 0 values, as shown in Figure 2.9, we can observe that both ΔD,opt and neff,opt increase monotonically with εG 0 . Inspection of the explicit definition of threshold gain helps explain the observed behavior in terms of the electric field E:

00 εGth

ð∞ ð 00 00 jEðρÞj2 ρdρ εM jEj2 dA εM Rtotal Metal ¼ ð ¼ ð Rcore 2 jEj dA jEðρÞj2 ρdρ Gain

ð2:3Þ

0

where the second equality is introduced for modes with azimuthal symmetry, such as the TE01 mode under consideration. Allowing ourselves the heuristic assumption that the electric field inside a bulk polarized material with relative permittivity εR is reduced from its free-space value by a factor of εR, then according to Equation (2.3) we would

46

Photonic Mode Metal-dielectric-metal–based Nanolasers

Figure 2.9

ΔD,opt and neff,opt as functions of εG 0 . εM, εD, λ0, and Rtotal are fixed at, −130 − j3.0, 2.16, 1.55 μm, and 0.45 μm, respectively. Reprinted from reference [102] with permission from Optical Society of America (OSA).

anticipate that εGth00 is proportional to εM00 (εG 0 )2/(εM 0 )2. However, Figure 2.7 shows that εGth00 increases with a decreasing εG 0 . Obviously, the problem in hand does not consist of a bulk polarized medium, so we modify our heuristic approach by incorporating neff into the proportionality. Namely, by studying the results of Figure 2.9, we observe that εGth00 is roughly proportional to εG 0 2 if we reduce εG 0 by the factor neff2 to account for the guided nature of the mode inside the CGW. Hence an approximate expression to Equation (2.3) can be posited:  h   i 2 2 00 00 εGth;opt ffi εM ε0G = ε0M nopt eff

ð2:4Þ

In Figure 2.10, εGth,opt00 is plotted according to Equation (2.4) as a function of εG 0 , along with the numerical solution to Equation (2.3). We observe that Equation (2.4) approximates the numerical solution to within a factor of two for all εG 0 . Furthermore, if εGth00 and neff are substituted for εGth,opt00 and neff,opt, then Equation (2.4) approximates the numerical solution to Equation (2.3) within a factor of two for all Rcore>300 nm. Figure 2.11 shows the percentage error in Equation (2.4) with this substitution over the range of Rcore and εG 0 values used in Figure 2.8 to Figure 2.10. The error is defined as 100|εGth,N00 –εGth,A00 |/εGth,N00 , where the subscripts N and A refer to numerical and analytical, respectively. We observe that in the region of most interest to the designer, that is, near Rcore,opt, the error is quite low, while it increases rapidly for smaller Rcore due to the more rapid variation of neff with decreasing Rcore, per Figure 2.8. Admittedly, Equation (2.4) is not rigorously derived; however, it clearly holds value as a design tool.

2.2 Invariance of Optimal Metallo-dielectric Waveguide Geometry

47

Figure 2.10

εGth,opt00 as function of εG 0 , (open squares) numerical solutions of Equation (2.3) and (solid line) analytical approximation of Equation (2.4). εM, εD, λ0, and Rtotal are fixed at −130 − j3.0, 2.16, 1.55 μm, and 0.45 μm, respectively. Reprinted from reference [102] with permission from Optical Society of America (OSA).

Figure 2.11

Error in Equation (2.4) with εGth00 and neff substituted for εGth,opt00 and neff,opt, relative to numerical solution of Equation (2.3) as function of Rcore with εG 0 parameterized. εD, εM, λ0, and Rtotal are fixed at 2.16, −130 − j3.0, 1.55 μm, and 0.45 μm, respectively. Reprinted from reference [102] with permission from Optical Society of America (OSA).

48

Photonic Mode Metal-dielectric-metal–based Nanolasers

We can summarize the nonintuitive main result of this letter in the following manner. One can begin with a material-geometry selection and solve the original eigenvalue problem by obtaining the zeros in the (εG00 , βz 0 ) plane. This process is then continued, varying Rcore with Rtotal fixed, until a minimum threshold gain, εGth,opt00 , and the corresponding Rcore,opt and neff,opt are found. Next, the metal permittivity is changed. The zeros in the (εG00 , βz 0 ) plane necessarily shift. However, by maintaining the imposed threshold condition, neff00 = 2πβz00 /λ0 = 0, εG00 is forced to respond proportionally to changes in εM00 . Because neff 0 remains unchanged or changes very slightly, the electric field distribution in the CGW remains unchanged or changes very slightly. Prior to changing εM, the electric field distribution was such that εGth00 = εGth,opt00 , and so it follows that the new threshold gain is also an optimum. The virtual lack of change of neff 0 and the constraint that neff00 = 0 are sufficient conditions for the invariance and weak dependence of ΔD,opt (equivalently, Rcore,opt) with respect to εM00 and εM 0 , respectively. The invariance and weak dependence of ΔD,opt on εM00 and εM 0 implies that once an optimized geometry is found for a given set of εG 0 , εD, and λ0, different metals may be used without affecting the numerical results. For example, if laser cavities employing silver cladding are rigorously designed, they need not be re-designed if the fabrication process necessitates the use of gold or aluminum claddings. Furthermore, data obtained from executing the optimization procedure over a wide geometric parameter space may be used in the development of approximate analytical expressions to expedite the design process. Based on the preceding results, approximations to ΔD,opt may be applied to structures with an arbitrary metal-cladding. The optimal shield thickness ΔD,opt is nearly a linear function of Rtotal, as seen in Figure 2.12. An approximation to the numerical solution that describes this relation is Δ,D,opt~(0.71Rtotal – 0.14) μm, which is accurate to within 2.5% over the range of Rtotal values from 0.30 μm to 0.70 μm. For the range of Rtotal values above 0.4 5μm, a better fit is ΔD,opt~(0.74Rtotal – 0.16) μm, which is accurate to within 1%. While the value of εGth,opt00 does depend on εM00 , we may still approximate it in a similar fashion. Figure 2.12 also plots the numerical solution of log10(εGth,opt00 ) versus Rtotal on a linear scale. Clearly, the logarithm of εGth,opt00 is almost inversely proportional to Rtotal. An approximation expressing this fact, log10(εGth,opt00 ) = (−7Rtotal + 0.064) μm, is also plotted and is accurate to within 10% over the range of Rtotal values from 0.35 μm to 0.75 μm. Thus, one may approximate εGth,opt00 explicitly in terms of the material parameters via Equation (2.4), or implicitly through the total radius via a logarithmic approximation.

2.3

Metallo-dielectric Nanolaser Fabrication The schematic of the open 3D laser resonator is depicted in Figure 2.13, in which a gain core is suspended in a bilayer shell of SiO2 and metal. While the design of Figure 2.2(c) uses a lower plug of SiO2, InGaAsP gain material is typically grown on InP substrate in standard III-V semiconductor manufacturing. Because of InP’s higher refractive index

2.3 Metallo-dielectric Nanolaser Fabrication

49

Table 2.1 Multiple quantum well InGaAsP/InP epitaxial structure. Name

Material

Thickness(A)

Number of layers

Capping layer Active region

InP 1.3 Q InxGa1−x AsyP1−y 1.6 Q InxGa1−x AsyP1−y 1.3 Q InxGa1−x AsyP1−y InP

100 300 100 200 ~325 micron

1 1 15 15

Substrate

Figure 2.12

ΔD,opt and log10(εGth,opt00 ) as functions of Rtotal. The solid triangles correspond to numerical solutions of Equation (2.3), whereas lines correspond to linear approximations for ΔD,opt and log10(εGth,opt00 ), respectively. εM, εD, εG 0 , and λ0 are fixed at −130 − j3.0, 2.16, 11.56, and 1.55 μm, respectively. Reprinted from reference [102] with permission from Optical Society of America (OSA).

(n = 2.1) than SiO2’s (n = 1.46), the cutoff behavior in the lower plug region will suffer. To combat this problem, air can be utilized as the lower plug material (Figure 2.13). The air plug provides even better confinement than the SiO2 plug, albeit at the cost of the possible gain medium degradation resulting from its exposure to air. The device depicted in Figure 2.13 is pumped through the bottom aperture of the air plug, and the emitted light is collected from the same aperture. InGaAsP MQW is chosen as the active material, because it has more material gain than bulk InGaAsP, with all other conditions equal. To obtain a gain cylinder height of 480 nm, 15-period MQW is chosen, with In0.564Ga0.436As0.933P0.067 (1.6Q) well layers of length LW = 10 nm and In0.737Ga0.263As0.569P0.431 (1.3Q) barrier layers of length LB = 20 nm. An additional top barrier layer of 30 nm makes the total active region height hcore = 480 nm. The material stack composition of the semiconductor stack is summarized in Table 2.1.

50

Photonic Mode Metal-dielectric-metal–based Nanolasers

Figure 2.13

Schematic view of a practical realization of the laser cavity, compatible with planar fabrication techniques. The air gap at the bottom of the laser is formed after selective etch of the InP substrate. In the designed cavity, the values for h1, h2, and h3 are 200 nm, 550 nm, and 250 nm, respectively. Reprinted from reference [107] with permission from Optical Society of America (OSA).

Although one typically designs for cylindrical cavities, the fabricated structure would always be elliptical, due to fabrication imperfections. To reflect the realistic structure, it is therefore helpful to replace the gain core radius rcore with rmajor and rminor, denoting radii of major and minor axes, respectively, to emphasize the ellipticity. For example, in a fabricated structure, in accordance with notations in Figure 2.13, rmajor = 245 nm and rminor = 210 nm (when the designed rcore = 230 nm); h1, h2, and h3 are 200 nm, 550 nm, and 250 nm, respectively. The SiO2 shield layer has thickness ΔD ≈ 200 nm. Another practical concern is the choice of the metal-cladding. While the design of Figure 2.2 uses gold as the cavity metal, in practice, the low adhesion of gold to SiO2 causes separation of the dielectric portion of the structure from the metal layer (during the creation of the air plug, as described in the fabrication at the end of this section). Fortunately, aluminum exhibits better adhesion properties, and at the near-infrared wavelength of interest, its optical properties are very close to gold. (The cavity Q of the resonator with rmajor = rminor = rcore = 230 nm using an aluminum coating (ε = −95.9 – j11) [101] is 1004, which is on a par with 1030 for gold. Assuming the aluminum cladding thickness to be 70 nm (twice the skin depth), the height and the major and minor total diameters of this laser are 1.35 μm, 1.03 μm, and 0.96 μm, respectively, resulting in a laser cavity that is smaller than its emission wavelength in all three dimensions. The cross sections of |E| for the target TE012 mode are depicted in Figure 2.14, using rmajor = rminor = rcore = 230 nm in numerical simulations.

2.4 Optical Pump Penetration Analysis

Figure 2.14

51

Cross sections of |E| for the TE012 mode of the cavity. Reprinted from reference [35] with permission from Macmillan Publishers Ltd.

The metallo-dielectric laser structure is fabricated from an InGaAsP MQW gain layer grown on InP (material composition is detailed in Table 2.1). Hydrogen silsesquioxane (HSQ) negative tone electron-beam resist is patterned into arrays of dots (Figure 2.15(a)) using electron-beam lithography, and the size of the dots is varied by changing the pattern size and/or the electron-beam dos age. Cylindrical structures are then etched using CH4/H2/Ar reactive ion etching (RIE) (Figure 2.15(b)). Using an optimized and calibrated plasma-enhanced chemical vapor deposition (PECVD) process, the SiO2 shield layer is grown to a thickness of ~200 nm (Figure 2.15(c)). Note that the outline of the embedded gain core is visible through the SiO2 layer. A layer of aluminum with a minimal thickness of 70 nm is then sputtered over the SiO2 covered pillars (Figure 2.15(d)). The sample is then bonded on the upper side to a glass slide using SU-8, and the InP substrate is subsequently removed in a selective HCl etch, leaving an air void under the structure. Figure 2.15(e) shows the tilted bottom view of an air void, with the lower face of the gain core visible inside. Figure 2.15(f) shows the normal bottom view (with enhanced contrast levels) of a similar void. The faint outline of the SiO2 shield is discernible in this image, verifying the 200 nm thickness of the shield.

2.4

Optical Pump Penetration Analysis For nanolasers that are optically pumped, pump penetration into the core is an important factor to be considered, given the small size of the input aperture. For geometries similar to that shown in Figure 2.13, to estimate the amount of pump power absorbed by the core, full 3D finite element analysis can be carried out over a range of core sizes (with fixed dielectric shield thickness), assuming illumination with a plane wave at a pump wavelength of 1064 nm and a peak illumination intensity of 1 KW/mm2. The result is

52

Photonic Mode Metal-dielectric-metal–based Nanolasers

Figure 2.15

Various stages of the fabrication process: (a) Array of e-beam patterned HSQ resist dots. (b) RIE etched pillar after oxygen plasma and BOE cleaning. The faint bump in the middle indicates the boundary between the InGaAsP and InP layers. (c) Etched pillar after PECVD of SiO2. The outline of the semiconductor pillar can be seen through the silica layer. (d) Silica covered pillar after undergoing aluminum sputtering (70 nm). (e) Tilted bottom view of one of the samples after selective InP etch with HCl. The surface is composed of the PECVD deposited SiO2. (f) Contrast enhanced normal bottom view of a cavity. The circular outline around the air hole is due to the dielectric shield and agrees well with the target dielectric shield thickness of 200 nm. Reprinted from reference [35] with permission from Macmillan Publishers Ltd.

shown in Figure 2.16. Looking at the absorbed power as a function of gain core radius, the solid curve in Figure 2.16(a) shows that the total power absorbed in the core exhibits oscillations at small core sizes, which flatten out as the core size increases. These oscillatory features are also present when a perfect conductor is substituted for the aluminum layer (Figure 2.16(a), dashed curve), eliminating the possibility that this phenomenon is a manifestation of surface plasmon related effects (e.g., extraordinary transmission through a metallic aperture [108]). Rather, it indicates that the oscillatory behavior is due to simple resonance of the pump inside the metallic cavity (which is stronger for smaller core sizes, since a smaller proportion of the core is absorptive). Interestingly, for smaller core sizes a significant proportion of the pump power is funneled through the silica layer and absorbed through the sidewall of the gain core (Figure 2.16(b)), which is an indirect benefit of using the shield layer. The curve showing carrier density as a function of the gain core radius in Figure 2.16(a) shows the generated carrier density for a peak pump intensity of 700 W/mm2 and a (rectangular) 12 ns pump pulse width. Estimation of the carrier density can be performed using the rate equation dNc ðtÞ=dt ¼ ϕi ðtÞ  Bo Nc2 ðtÞ, assuming a carrier dependent recombination lifetime

2.4 Optical Pump Penetration Analysis

Figure 2.16

53

(a) Results of 3D FEM simulations showing the absorbed pump power in the gain core as the size of the core is varied. The core height is 480 nm and the solid and dashed curves correspond to aluminum and perfect conductor metal shields, respectively. The pump wavelength is 1064 nm, polarized in the x direction, and the incident intensity is assumed to be 1 KW/mm2. The red curve shows the estimated threshold carrier density assuming a 12 ns pulsed. The pump power is assumed to be the threshold value of 700 W/mm2. (b) Pump power flow (arrows) showing how the dielectric shield funnels the pump beam through the sides of the gain disk. The core radius in this case is 116 nm, corresponding to the first absorption peak for an aluminum shell. Reprinted from reference [35] with permission from Macmillan Publishers Ltd.

τr ¼ ðBo Nc Þ1 , where Nc is the carrier density and Bo ¼ 1010 cm3  s1 [109]. ϕi ðtÞ is the incoming photon flux signal, calculated from the pump peak power. The refractive index drop due to carrier effects can be estimated using results derived in reference [110]. Depending on the carrier density level, the effects of band filling, bandgap shrinkage, and free carrier absorption can induce a positive or negative refractive index change, depending on the wavelength and the dominating process(es). For InGaAsP at 1.55 μm and for the high carrier densities estimated in our case (about 1.2×1019 cm−3 for a core diameter of 520 nm), the combination of band filling and free carrier absorption dominates the bandgap shrinkage effect, resulting in an estimated net drop of approximately 0.1 in the refractive index [110]. Another phenomenon that may contribute to refractive index decrease is the compressive pressure exerted by the sputtered aluminum shield after it cools down to room temperature. Even though a 70-nm-thick aluminum layer is sufficient to create optical confinement, usually, the sample would be covered with additional aluminum (on the order of one micron) for better heat sinking. This metal layer will have a larger thermal contraction than the SiO2/InGaAsP core and will exert compressive pressure on it, which will result in a drop in the InGaAsP refractive index. However, this effect is likely to be negligible in these structures. As the device cools, the difference in thermal contraction (24 ppm/celcius for aluminum vs. 5 ppm/celcius for both SiO2 and InGaAsP) is accommodated partly by compression of SiO2/InGaAsP core, and partly by compression of nearby aluminum. Even if all of the compression occurred in the core, it would amount to just 0.0019 for a

54

Photonic Mode Metal-dielectric-metal–based Nanolasers

100 celcius drop in temperature. Assuming the Young modulus of the core to be nominally E = 70 GPa, the stress corresponding to this small compression would be less than 0.14 GPa, and the effect of this small stress on the refractive index less than 0.004 [111].

2.5

Metallo-dielectric Nanolasers on Silicon Realization of silicon-compatible active optical components is critical for creating integrated silicon photonic circuits, and a major step toward their integration with complementary metal-oxide semiconductor (CMOS) compatible platforms. Silicon has good thermoconductive properties, high-quality oxide, and low cost and is available in high-quality wafers, making it invaluable for integrated electrical circuits and waveguides. Unfortunately, the indirect bandgap of silicon poses a fundamental barrier to enabling light amplification and stimulated emission in this material. Successful advances toward overcoming this limitation include demonstration of silicon Raman lasers [112] and light emission in silicon-based nano-engineered materials [113]. However, Raman laser operation is based on optical scattering, which makes it essentially limited to operation under external optical pumping. Silicon nanostructures, in turn, suffer from low gain and, therefore, low efficiency. An alternative solution is to build hybrid optical devices through integration of III-V gain with silicon [114] using wafer bonding. III-V semiconductor compounds offer the benefits of a direct energy bandgap with wide achievable range, and high carrier mobility. This makes the III-V material system optimal for active optical elements and logical devices such as lasers, switches, and modulators. As discussed in Section 2.1, although the design of Figure 2.2(c) uses a lower plug of SiO2, in standard semiconductor growth, the InGaAsP gain region is grown on top of III-V semiconductors, in this case, InP substrate (Table 2.1). As a result, an air plug was employed by removing the InP substrate beneath the InGaAsP gain region in the cavity. In the design depicted in Figure 2.13, the nanolaser is optically pumped and mounted on a glass slide with the output light propagating in free space (illustrated again in Figure 2.17(a)). However, this realization is not only prone to material/device degradation due to gain medium’s exposure to air but also not compatible with chip-scale integration applications if these nanolasers are to be used as on-chip sources. The wafer-bonding approach offers a path to the realization of the nanolaser on a silicon-compatible platform (illustrated in Figure 2.17(b)), and the possibility of the realization of an electrically pumped nanolaser on silicon. Because the nanolaser design of Figure 2.2(c) utilizes low refractive index SiO2 plug regions, the goal is to employ a low-index plug material in a silicon-compatible platform. To achieve this, one can mediate III-V-to-silicon wafer bonding with SiO2 (Figure 2.17(b)). The mediator layer can be chosen to be thick enough to mimic the 550-nm-long air plug of Figure 2.13. Next, a reliable and versatile wafer-bonding method, which could be used for both III-V-to-Si and III-V-to-SiO2/Si integration, needs to be selected. There are a wide variety of available wafer-bonding techniques, and they can be roughly divided into

2.5 Metallo-dielectric Nanolasers on Silicon

(a)

Optical pump Free space

Laser emission

MQW SiO2 Metal Polymer Glass slide Figure 2.17

55

(b) Metal SiO2 MQW

Optical pump

Laser emission

Silicon substrate

(a) Scheme of the original subwavelength metallo-dielectric laser setup; (b) scheme of a silicon-compatible wafer-bonded metallo-dielectric laser. Reprinted from reference [115] with permission from Institute of Electrical and Electronics Engineers (IEEE).

two broad categories: direct and indirect [116]. Direct methods rely on molecular forces between the two materials. The main advantage of direct methods is the immediate vertical proximity between the III-V and silicon layers, which allows the composite structure to be treated as a single wafer during fabrication and hybrid device design. Atomic flatness and pristine condition of both surfaces are imperative for direct methods to work [117]. Pressure and/or thermal treatment are usually applied to assist molecularlevel interactions. Hydrophilic bonding and hydrophobic bonding are examples of the direct bonding technique. In hydrophobic bonding, all native oxides must be stripped to leave both surfaces hydrogenated, and bonding is carried out in a vacuum, under pressure at room or elevated temperatures. Hydrophilic bonding relies on Van der Waals forces to facilitate attraction between the surface species. Either wet chemistry or plasma treatment can be used to make the surfaces hydrophilic. After Van der Waals bonding, the composite structure typically undergoes thermal processing to encourage covalent bond formation. Indirect methods use a third mediating material between the materials to be bonded. Metal-based anodic and adhesive bonding methods are all indirect methods. The indirect approach helps circumvent the problems of lattice mismatch, surface roughness, and poor planarity. Metal-based bonding techniques are perhaps the most established waferbonding techniques to date. The oldest technique, thermos-compression bonding, relies on atom diffusion between different metals at elevated temperatures. Another popular indirect method, eutectic bonding, is similar to soldering and is based on metal alloy formation. Eutectic bonding uses lower temperatures than thermos-compression but may have poor bond stability. It is also incompatible with CMOS processing because of ion diffusion. On the other hand, anodic bonding is normally used for bonding semiconductors to borosilicate glasses. Strictly speaking, anodic bonding does not require an extra material layer, but one of the materials (typically, the glass cathode) must be doped with high-mobility charge carriers. This can have an adverse effect on electronic device performance and, thus, is not a good choice for CMOS integration. Some of the popular wafer-bonding adhesives include bensocyclobutene (BCB), SU-8 polymers, and HSQ.

56

Photonic Mode Metal-dielectric-metal–based Nanolasers

Typically, adhesive wafer bonding is carried out at lower temperatures than most other wafer-bonding techniques, but the downside is low temperature stability of the bonding interface and poor surface quality. In the case of silicon-compatible nanolasers operating in optical communication (near infrared) frequency range, the choice is determined by the gain material. To bond InGaAsP MQW active layer to a silicon base, a direct low temperature (below 400°C) bond is required. Conventional high-temperature direct wafer bonding (fusion bonding) is known to have an adverse effect on the III-V layer quality due to a thermal expansion constant mismatch of materials to be bonded. Some of the available lowtemperature wafer-bonding methods may be useful but would require additional studies of the bonding mediator’s optical and material properties (adhesive wafer bonding), its possible chemical interaction with the gain layer (anodic wafer bonding), and incorporation of these new layers into the original nanolaser model. Another direct, low-temperature wafer-bonding method is plasma-assisted wafer bonding. It was first reported by Liang et al. [118, 119], in which the method was used to realize silicon-on-insulator (SOI)–compatible optoelectronic devices [114], and this approach is most applicable for integrating III-V–based nanolasers on silicon photonics platforms. Using the composite gain-metallo-dielectric for the InGaAsP/InP material system but wafer bonded to SiO2/silicon (silicon wafer with thermally grown SiO2 layer), Figure 2.18(a), similar to the conceptual drawing of Figure 2.17(b), shows a schematic of the envisioned device. This nanolaser is optimized for minimum lasing threshold at a nominal wavelength of 1550 nm and outer radius of the resonator Rout = 460 nm. For a structure of these dimensions, the optimal shield thickness (Δ) and gain core radius (Rcore) are 200 nm and 250 nm, respectively, similar to geometrical parameters used in Figure 2.13 [117]. Because of the omission of the InP substrate etch that is used to create the air plug as in Figure 2.13, the poor adhesion between low-loss metal (such as

Figure 2.18

(a) Schematic drawing of the wafer-bonded metallo-dielectric resonator design; (b) FEM simulation of TE021 mode in metallo-dielectric resonator with Rout = 460 nm, optimized for 1550 nm wavelength. Reprinted from reference [115] with permission from Institute of Electrical and Electronics Engineers (IEEE).

2.5 Metallo-dielectric Nanolasers on Silicon

Figure 2.19

57

Fabrication steps of a wafer-bonded metallo-dielectric laser: (a) wafer bonding of InGaAsP to SiO2/Si; (b) resultant InP/InGaAsP/SiO2/Si structure; (c) wet etching of InP substrate and the optical microscope image of 500 nm InGaAsP MQW layer bonded to SiO2/Si chip; (d) e-beam lithography patterning; (e) laser pillar after two-step RIE of InGaAsP and SiO2; (f) PECVD of SiO2 shield; (g) metal sputtering (silver). Reprinted from reference [115] with permission from Institute of Electrical and Electronics Engineers (IEEE).

silver and gold) with the dielectric shield material is no longer an issue. This is another benefit that results from the wafer-bonded material platform. Silver can therefore be chosen as the cavity metal encapsulating the nanolaser. Because the refractive index of SiO2 is only slightly higher than that of air (1.46 versus unity), the replacement of the air plug with a SiO2 one does not introduce any significant changes in the mode profile, which is evident by comparing Figure 2.18(b) with Figure 2.14. The wafer-bonded subwavelength metallo-dielectric structures are fabricated from an InGaAsP multiple quantum well gain layer, bonded to a silicon wafer with a SiO2 layer on top. The complete fabrication scheme is outlined in Figure 2.19. The silicon wafers have a 6 μm thermally grown oxide layer on the surface. The MQW InGaAsP wafer is epitaxially grown on an InP substrate (material stack detailed in Table 2.1). InGaAsP MQW active layer bonding to the SiO2/Si was accomplished through low-temperature plasma-assisted wafer bonding (Figure 2.19(a–c)). This wafer-bonding technique is typically performed for an InGaAsP/InP chip of ~ 1 cm2 area and a SiO2/Si chip of ~ 2 cm2 area. The low-temperature bonding starts with ultrasonication of the cleaved wafers in solvents to remove all particles that would inhibit interaction between the surfaces. InGaAsP/InP is dipped in HCl for 10 seconds prior to the wafer bonding to remove the InP capping layer.

58

Photonic Mode Metal-dielectric-metal–based Nanolasers

Ultrasonication in acetone and isopropanol (IPA), followed by thorough deionized water rinse are necessary to eliminate particles from the bonding surfaces. This is a crucial step, because a single 1 µm size particle can result in an un-bonded area of 1 cm2 [117]. Next, the surfaces are to be stripped of native oxides and organic and ionic contaminations. The chemical treatment is performed using standard RCA clean (NH4OH:H2O2:H2O = 1:2:10) for 10 minutes at 65–75°C on the InGaAsP/ InP chip, and modified RCA clean (HCl:H2O2:H2O = 0.2:1:5) for 10 minutes at 80°C on the SiO2/Si chip. Then, the InGaAsP/InP chip is immersed into NH4OH to remove any native oxides and small contaminations left from the previous step, while SiO2/Si is treated in H2SO4:H2O (3:1) for 10 minutes to strip any traces of organic contaminants. The chemical clean is followed by ultrasonication in acetone, IPA, and careful DI water rinse and drying in N2 flow. After cleaning, both chips undergo O2 plasma surface activation in an RIE chamber with 20mTorr pressure, 30sccm O2 flux, and 50 W RF power for 45 seconds (typical conditions). The oxygen plasma treatment is immediately followed by a brief rinse in DI water to passivate the active surfaces with hydroxyl (–OH) groups. The water flow also takes away any new particles that may have accumulated during the plasma activation. Next, nitrogen flow-dried chips are manually mated. The Van der Waals force between the –OH groups promotes spontaneous mating of activated surfaces. The pair is annealed for 17 hours at 300°C in an oven to form strong covalent bonding and encourage out-diffusion of the H2O and H2 by-products through the thick SiO2 layer from the InGaAsP/SiO2 interface. Lastly, the InP carrier substrate is selectively etched by HCl from the bonded sample to obtain the composite InGaAsP/SiO2/Si structure (Figure 2.19(c)). After bonding III-V wafer to SiO2/Si, the next step is to define the nanolaser cavity. Similar to the version with the air plug, e-beam lithography can be performed on HSQ resist to create a mask (Figure 2.19(d)). This is followed by the two-step RIE to form the cylindrical gain core and bottom SiO2 plug. Note that the RIE has to be carried out on the gain layer and the SiO2 “plug” layer separately, using appropriate etching chemistry for each of the two materials. First, the InGaAsP MQW layer is dry etched in CH4/H2/Ar chemistry. Next, the SiO2 layer undergoes CHF3/Ar RIE to obtain a 500 nm tall SiO2 post (Figure 2.19(e) depicts a schematic drawing of the structure after both these steps are performed). The HSQ mask is removed during the last fabrication step along with SiO2, since its chemical composition is quite similar to that of SiO2. The sample is then treated in microwave oxygen plasma to eliminate polymer buildup. A scanning electron microscope (SEM) image of a sample laser after the cleaning is presented in Figure 2.20(a). Next, 200-nm-thick PECVD SiO2 is carried out to form the low-index shield around the InGaAsP core (Figure 2.19(f)). The corresponding SEM image is shown on Figure 2.20(b). In the final step, silver is conformally sputtered around the pillars to complete a laser cavity (Figure 2.19(g)). To conclude this section, we remark that this approach to realize subwavelength scale coherent sources, combined with the latest III-V-to-Si wafer-bonding solutions, is a promising path for the realization of highly integrated and miniaturized silicon photonic light sources.

2.6 Micro-photoluminescence Characterization of Nanolasers

Figure 2.20

SEM pictures of (a) a laser pillar after two-step RIE; (b) the same pillar after PECVD of 200-nm-thick SiO2 layer. Reprinted from reference [115] with permission from Institute of Electrical and Electronics Engineers (IEEE).

2.6

Micro-photoluminescence Characterization of Nanolasers

59

One of the most common methods to characterize laser performance is the microphotoluminescence (micro-PL) optical characterization. Micro-PL measurements reveal active devices’ spectroscopic characteristics at different excitation levels, as well as light-in vs. light-out (LL curve) behaviors. While this characterization can be easily carried out with conventional electrically pumped semiconductor lasers that have ample output power, it is not the case with nanolasers. To date, nanolasers are typically optically pumped with pico- to nano-watt output power, making both excitation and detection a challenge in micro-PL measurements. In this section, we use metallodielectric nanolasers designed in Section 2.1 as devices under test, to show the construction of micro-PL setup for optically pumped nanolasers. We first consider the excitation part of the micro-PL. Given the design of the metalcladding and the bottom aperture, one has to work in reflection mode; that is, the same optics need to be used for optical pumping and for the collection of the emitted light. Furthermore, we’d have to be able to illuminate and image the sample effectively such that we know which laser is being tested, which means large magnification is required. Lastly, we need to perform spectral measurements with high resolution (sub nm) with very little signal (pico- to nano-watt peak power). These constraints lead to the design depicted in Figure 2.21. The sample is first illuminated by 1.55 µm super-luminescent light emitting diode (SLED) (resulting illuminated sample is shown Figure 2.22(a)). To reduce spatial coherence and provide uniform illumination, the SLED output is passed through a diffuser (resulting illuminated sample is shown Figure 2.22(b)). It is then imaged via two cascaded 4–f imaging systems (L1, L2, L5, and L6 in Figure 2.21) to an InGaAs CCD camera. The images in Figure 2.22 are obtained with a camera that has a resolution of 252 × 316 pixels, and 30 µm pixel size. This enables users to locate the laser device of interest and position it in the center of the field of view for the pumping and characterization.

60

Photonic Mode Metal-dielectric-metal–based Nanolasers

Figure 2.21

Schematic of micro-PL characterization setup.

Figure 2.22

Imaging of a nanolaser sample: (a) without the use of the diffuser, (b) with the diffuser, (c) nanolaser emitting light.

The nanolaser sample can be pumped using a 1064 nm pump laser. While CW pumping is ideal, pulsed pumping is usually used at early testing stages of laser cavity design for reduced heating. A combination of pulse width and repetition rate should be chosen such that high enough peak power is obtained to reach lasing threshold. In the meantime, the average power is low enough such that thermal damage of the device under test is avoided, but it is sufficiently high to provide sufficient signal-to-noise ratio in detection. To measure, the pump beam is delivered through the dichroic beam splitter into a microscope objective (L1 in Figure 2.21) and focused down on the sample to a spot size

2.6 Micro-photoluminescence Characterization of Nanolasers

61

of ≈ 8×8 µm in this example. The objective is also used to collect the emitted light. Because the pump is at 1064 nm and the light emitted by the laser is around 1500 nm, this presents us with the problem of chromatic aberration. Namely, the focal plane is not at the same location for the two wavelengths. To combat the chromatic aberration, a telescope can be introduced (lenses L7 and L8 in Figure 2.21). With the telescope, the divergence of the pump beam can be adjusted such that the focal planes of the two wavelengths coincide. To collect the emitted light by the nanolaser, a high-pass edge filter is used to remove the residual power from the pump laser. Figure 2.22(c) shows nanolaser emission on the CCD camera. At this point there are a few options on how to characterize the light output from the nanolaser, each with its advantages and disadvantages. We can do one of three things: – By removing the mirror M1, and leaving mirror M2 in place, we can re-image the laser through the 4–f imaging system composed of lenses L3 and L4. We can then couple the collected nanolaser light to the multi-mode optical fiber and pass it to an InGaAs fixed-grating spectrometer, which gives information about the spectrum of the light. The advantage of this spectrometer is that the measurement is fast, but it comes at the expense of a large signal-to-noise ratio and, more importantly, a limited resolution. – Using the same mirror arrangement, we can couple the output of the nanolaser to a single-pixel thermoelectrically cooled InGaAs detector that gives us the total optical power emitted by the nanolaser. This is a way to gather quick data on the laser output, but it does not contain any spectral information. – Last, by removing both M1 and M2 mirrors, we can pass the collected nanolaser light to the monochromator, which gives excellent spectral resolution, down to 70 pm. The disadvantages of this approach is that monochromator by definition analyzes one wavelength at the time, thus making the measurement rather slow. Because the light output of the metallo-dielectric nanolasers is on the order of few nanowatts, lock-in detection is necessary to boost the signal-to-noise ratio. Out of the previous options, the highest-quality emission spectra are obtained using a monochromator equipped with a cooled InGaAs detector in a lock-in detection configuration. For lock-in detection, one needs to find an adequate modulated signal to which the lock-in amplifier locks. Although one could in theory use the individual pulses from the pulsed pump laser as a source of modulation, low noise detectors typically have bandwidth of a few KHz and therefore lack the bandwidth to utilize a high repetition rate from the pump laser (the detectors would see a high repetition rate pulse train as a CW signal). As a result, a low frequency mechanical chopper with frequency of around 1KHz can serve as a source of modulation. Under this arrangement, a sub picowatts signal from the nanolaser can be detected and spectrally characterized. Another option for detection is to use an arrayed detector, such as a CCD camera in conjunction with a spectrograph. Last, we note that the optical pumping efficiency can be increased by making the cavity resonant with the pump light [120]. Next, we look at emission characteristics of the nanolaser whose dimension is described in Figure 2.13. In this particular experiment, a pulsed laser emitting at

62

Photonic Mode Metal-dielectric-metal–based Nanolasers

Figure 2.23

Light-light curve for a nanolaser with major and minor core diameters of 490 nm and 420 nm (dotted curve). The same data set is shown as a log-log plot (dotted inset) together with the slopes for the PL, ASE, and lasing regions. Also shown are the images of the defocused emitted beam cross section (taken at about 10 μm away from the nanolaser exit aperture) for (I) CW pumping and (II) pulsed pumping. The appearance of the higher contrast fringes indicates increased coherence due to lasing. Reprinted from reference [35] with permission from Macmillan Publishers Ltd.

1064 nm with 12 ns pulses and 300 KHz repetition rate is used as the pump laser. The light-light curve of the nanolaser emitting at 1430 nm is shown in Figure 2.23 (dots), which shows a slope change indicating the onset of lasing at an external threshold pump intensity of about 700 W/mm2. The same data set is shown in a log-log plot (Figure 2.23, inset graph), with the slopes of different regions of operation indicated on the plot. The S-shaped curve clearly shows the transition from the photoluminescence (PL) to amplified spontaneous emission (ASE) and finally into the lasing regime. To compare the spatial coherence of the emitted light before and after the onset of lasing, the sample is defocused from the object plane of the imaging system by approximately 10 μm (away from the objective) and the resulting diffracted mode pattern is imaged onto the camera. The upper right inset of Figure 2.23 shows the emission patterns of the defocused laser image captured with the CCD camera, corresponding to CW (Figure 2.23-I) and pulsed (Figure 2.23-II) pumping situations. The average pump intensity in each case is approximately equal to 8 W/mm2. The transition

2.6 Micro-photoluminescence Characterization of Nanolasers

63

from PL (using a CW pump) to lasing (using a pulsed pump) is dramatically shown by the appearance of Airy-like patterns, which indicate the increase of emission coherence. Observing Figure 2.23-II, the defocused image forms a distinct spatial mode with increased fringe contrast, which is an indication of increased spatial coherence and is an evidence of lasing. The polarization of the emission has a strong linear component, which is due to the slight ellipticity of the gain core. The null feature at the center of the defocused pattern may be a result of a TE01 lasing mode (which also has a null in the center), but other effects (such as imaging aberrations) may also be the source of this feature. The polarization of the emitted beam is elliptical, with a large portion of the lasing power residing in one polarization. This does not agree with the expected azimuthal polarization of a TE01 lasing mode. The reason for this discrepancy is the slight ellipticity of the structures, which affects the emitted beam in two ways. First, the ellipticity splits the degenerate TE01 mode and results in a slightly elliptical lasing mode in the gain core section. Second, the elliptical metal-coated dielectric section that connects the gain core to free space acts as a weak polarizer and preferentially transmits one polarization, resulting in a rather strong linearly polarized component at the exit aperture of the laser. Only broad PL emission occurs in the CW case, owing to the low peak intensity. However, when the pump is switched to pulsed mode, lasing is achieved due to the 278fold increase in peak power. Figure 2.24(a) shows the evolution of the emission, from a broad PL spectrum to a pair of competing ASE peaks and finally into a narrow lasing line at 1430 nm. The measured linewidth of this particular laser is 0.9 nm.

(a)

(b)

Laser output (normalized)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1,300

Refractive index

3.6

3.2 3

1,400 1,500 1,600 Figure 2.24

3.4

194 640492 1,238 2,312

66

1,400

1,450 1,500 λ (nm)

1,550

(a) Evolution of the emission spectra from PL to lasing. (b) Effective refractive indices (dotted data points) of the pumped MQW gain medium at lasing wavelengths, back-calculated from lasing spectra obtained from an array of nanolasers. Error bars were calculated assuming ±5 nm error in measuring the disk diameters from the SEMs. The dashed curve shows the effective refractive index of the unpumped MQW layer, as measured by a Filmetrics interferometric analyzer. The solid curve is offset down from the dotted curve by a constant amount (0.102 RIU), which was chosen for best fit to the lasing data. The index reduction is consistent with the estimated free carrier effects. Reprinted from reference [35] with permission from Macmillan Publishers Ltd.

64

Photonic Mode Metal-dielectric-metal–based Nanolasers

Another way to verify the soundness and accuracy of the design and fabrication processes is to match the lasing wavelength with the target resonance of the cavity. However, owing to the high pump intensity, the refractive index of the gain core can vary substantially from its quiescent value, and this can considerably shift the lasing line from its target wavelength. Using the measured results from an array of lasers with slightly different sizes (which can be measured individually using a SEM) and exact 3D numerical simulation of each of the gain cores, the refractive index of the gain medium under pumping conditions was estimated at each lasing point (Figure 2.24(b), dots and error bars). Assuming a uniform drop over the spectrum of interest and using a least-squares fit of these data points, the estimated refractive index drop is ~0.102 refractive index units (RIU) (least-squares fit) lower than that reported by interferometric multilayer measurements of the unpatterned wafers under low illumination intensity (Figure 2.24(b), dashed line). This shift is attributed mainly to free carrier effects (a combination of band filling, bandgap shrinkage, and free carrier absorption) whose net effect at the estimated carrier density (about 1.2 × 1019 cm−3 for a 520 nm diameter core) is expected to cause a refractive index drop of about 0.1 RIU. Also a slight additional contribution (at most −0.004 RIU) may also be present due to compressive pressure on the gain cores, which is exerted by the thermal shrinkage of the aluminum layer after deposition in the sputtering chamber.

3

Purcell Effect and the Evaluation of Purcell and Spontaneous Emission Factors As the cavity size is reduced with respect to the emission wavelength, interesting physical phenomena, unique to electromagnetic cavities, arise. The cavity quantum electrodynamics (QED) effects caused by the interaction of the matter and electromagnetic field in subwavelength structures have been the subject of intense research in recent years [107, 121, 122]. The generation of coherent radiation in nanostructures has attracted considerable interest, owing both to the QED effects that emerge in small volumes and the potential of these devices in future applications, ranging from on-chip optical communication to ultrahigh resolution and high throughput imaging/sensing/ spectroscopy. One of the QED phenomena that is relevant in nanolasers is the Purcell effect. Purcell effect describes the enhancement or inhibition of the spontaneous emission rate of an emitter in a cavity compared to emission in free space [3]. In nanolasers, enhanced emission together with a reduced number of cavity modes relative to large lasers can have significant effects, especially on subthreshold behavior. These effects are generally desirable, as they tend to increase the utilization of spontaneous emission into the lasing mode and, consequently, lower the lasing threshold. The utilization of spontaneous emission into the lasing mode is termed spontaneous emission factor, β, which is defined as the ratio of spontaneous emission channeled into the lasing mode versus all modes. β may be considered a measure of the efficiency of the below-threshold carrier-photon dynamics and therefore has been seen as a figure-of-merit for nanolasers [5, 6]. Improving the Purcell factor Fp and the spontaneous emission factor β in nanolasers has attracted significant attention, largely motivated by the quest for energy-efficient operations in nano-scale devices. While the concept of thresholdless operation continues to be a subject of debate [15, 16], the modulation and efficiency improvements enabled by subwavelength and wavelength scale cavities, which are directly related to the Purcell effect in these cavities, are fairly well understood [8–10]. For example, with proper design, the cavity of a subwavelength laser may be designed such that most of the spontaneous emission is channeled into the lasing mode. In so doing, unwanted emission into non-lasing modes is mitigated, and the below-threshold efficiency is limited only by non-radiative recombination. If the desired cavity mode has the highest Purcell factor among all cavity modes, a high-β laser can be realized even in a multi-mode cavity. With this design goal in mind, it is first and foremost important to accurately evaluate the Fp of all cavity modes, taking into account the emitter (electron-hole pair in the case of semiconductor) environment and the gain material properties.

66

Purcell Effect and the Evaluation of Purcell and Spontaneous Emission Factors

For a cavity mode of quality factor Q, active region volume Va, and refractive index n, emitting at the free-space wavelength λ, the most commonly used Purcell factor expression, as originally described by Purcell [3], is "

# τbulk 3 Q λ 3 Fp ≡ ¼ ð3:1Þ 4π2 Va n τcav Equation (3.1) describes the maximum attainable value of the Purcell factor, under the assumptions that the cavity mode is resonant with the emitter transition, and that the emitter is located at an antinode (i.e., maximum field strength) of the cavity mode. Since the original description by Purcell, the Purcell effect has been studied in a number of general physical contexts, such as when the emitter and cavity mode are not on resonance [7, 123], when the spectral broadening of the emitter and cavity mode is comparable [9, 10, 124], and when the emitters are a collection of nonidentical quantum dots (QDs) [125]. While the original Purcell effect expression shown in Equation (3.1) considered radio frequency microcavities, the formal treatment of Purcell effect in the optical regime was not presented until the work of Gerard et al. [7]. Taking into account that the cavity mode frequency rarely exactly matches the transition frequency of the emitter, and that the spatial distribution of emitters follows the field distribution in the gain medium, Gerard et al. express Fp as " #" #

 #" 3 Q λ 3 jd  fðre Þj2 Δω2c Fp ¼ ð3:2Þ 4π2 Va n jdj2 4ðωe  ωc Þ2 þ Δω2c where d is the microscopic dipole; f (re) describes the electric field of the mode; and ωe, ωc, and Δωc are the emitter frequency, cavity frequency, and cavity full-width-at-halfmaximum (FWHM), respectively. In this chapter, we intend to outline a formal treatment of the Purcell effect in semiconductors nanolasers and include both inhomogeneous broadening (due to the distribution of carrier energies within the conduction and valence bands) and homogeneous broadening (due to intraband scattering) of semiconductors. We consider bulk or multiple-quantum-well (MQW) semiconductors as the active medium. The fundamental system in cavity-QED is a two-level emitter interacting with the electromagnetic field in a cavity [121, 126]. Characteristics of this system, such as the spontaneous decay rate, are not inherent to the emitter but depend on the interaction between the emitter and cavity modes. Further, the emitter-mode interactions undergo modifications as the cavity modes are modified by their environment, for example, the lossy boundaries of a nonideal cavity. One can use the emitter-field-reservoir model in the quantum theory of damping to treat such interplay. Within this model, the emitter-field interaction is modified to the extent that the field mode is modified by its environment. As will be shown in the ensuing section, all Purcell factor expressions currently in the literature are valid only in the hypothetical condition when the gain medium is replaced by a transparent medium [107]. Even under the transparent medium condition, the Purcell factor expressions given by Equations (3.1) and (3.2) are only valid if the cavity

3.1 Gain Medium and Its Excitation

67

lineshape is much wider than the gain medium inhomogeneous broadening lineshape. However, this is not the case in moderate- to high- Q semiconductor lasers. In this chapter, therefore, we consider both cavity and inhomogeneous broadening lineshapes and derive a general Purcell factor expression. We will see that in semiconductor nanolasers, the role of the cavity Q may be diminished, and Equations (3.1) and (3.2) may no longer adequately describe the cavity-modified spontaneous emission rate. We start by evaluating the properties of the gain medium, then develop a formal treatment for the Purcell effect. After evaluating the Purcell factor, we can then obtain another figure-of-merit in nanolasers – the spontaneous emission factor – and observe how these figures-of-merit are influenced by change in temperature.

3.1

Gain Medium and Its Excitation In thermodynamic equilibrium, all materials are absorbing; that is, the material gain of naturally existing materials is always negative. To excite materials such that they become amplifying requires the introduction of a pumping agent. Either an applied optical field (optical pumping) or an injection of electric current (electrical pumping) can pump the gain medium into a state of nonequilibrium, such that an optical wave propagating through the medium increases in amplitude. For example, the MDM metallo-dielectric nanolasers we studied in Chapter 2 operate under the optical pumping condition. The degree to which a semiconducting crystal is away from equilibrium may be described by the difference between the Fermi distributions of electrons in the conduction and valence bands. Assuming a two-band model, wherein optical emission occurs when an electron-hole pair recombines, the nonequilibrium of the material system increases as the probability of occupancy of the conduction band increases and the occupancy of the valence band decreases. In general, the Fermi distributions depend on the magnitude of pumping and change with changes in transition energy. All materials exhibit a pump-dependent gain spectrum, which according to the direct-transition model for bulk media [127], is K jMj2 ½ f2 ðℏωÞ  f1 ðℏωÞ ρr ðℏωÞ ð3:3Þ ω   where the constant K ¼ πe2 = cnr ε0 m20 , jMj2 is the mean-square momentum matrix element; the bracketed term, known as the inversion, is the difference between the conduction and valence band Fermi distributions; and the final term is the reduced density of states. The relationship between the pump − optical intensity or injected current density − and the inversion depends essentially on the quasi-Fermi levels. For optical pumping, the pump intensity, Ip (W∙m−2), is related to the carrier density Nc (cm−3) by gðℏωÞ ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eIp Ag ηT Nc ¼ Eph B2eff

ð3:4Þ

68

Purcell Effect and the Evaluation of Purcell and Spontaneous Emission Factors

where Ag is the cross-section area of the gain medium, ηT is the fraction of source photons that excites the electron-hole pairs, Eph is the average energy per pump photon, and Beff is an effective recombination parameter typically on the order of 10−10 cm3/s [14]. For current injection, the pump current ip (A) is related to the carrier density by Nc ¼

ip τηi eAg LQW

ð3:5Þ

where τ is the injected carrier lifetime, ηi is the fraction of injected carriers to free carriers in the medium, and LQW is the thickness of the quantum-well (QW) layer [15]. For a QW laser, considering only contributions from the first subband, the quasiFermi levels are ) ! #    ( " πℏ2 Nc LQW Fc Ec þ ð3:6Þ ¼ ln exp  1 kB T Fv Ev  2me kB T where Ec(v) is the energy at the conduction (valence) band edge, and zero doping is assumed [6]. Once the quasi-Fermi levels are determined, the gain spectrum can be computed numerically; results of this computation are shown in Figure 3.1. Clearly, higher carrier densities lead to higher gain at a given wavelength. The parameter plot of Figure 3.1(b) illustrates, however, that the increase of the material gain is sublinear at a certain point, which is known as saturation. A difference between the gain spectrum for QW systems, compared to bulk (refer to Appendix B.2), is that the peak gain always occurs at the lowest photon energy. The major difference – the reason QWs are often chosen over bulk − though is that the peak gain, in general, is greater for QW systems, due to the quantized density of states.

Figure 3.1

(a) Contour plot of material gain (cm−1) versus carrier density and photon energy, for a material system of 10 InGaAsP QWs. (b) Material gain versus wavelength of same system, with carrier density parameterized. The calculations are based on the direct-transition model.

3.2 Formulation of Purcell Effect in Semiconductor Nanolasers at Room Temperature

3.2

69

Formulation of Purcell Effect in Semiconductor Nanolasers at Room Temperature With an understanding of the semiconductor gain medium, we proceed to evaluate the Purcell effect in semiconductor nanolasers at room temperature. First, we apply the results from the nonrelativistic QED treatment of 2-level systems (summarized in Appendix A) to a 3-level laser, in which emitters are excited from the ground state |1> to an excited state |3> and quickly decay from state |3> to a lower state |2>; the lasing transition is between states |2> and |1> [107]. Semiconductor lasers in particular are frequently modeled in this manner, even though their underlying physics differs: state | 2> describes the condition where a conduction band state is occupied and the valence band state of the same crystal momentum is vacant, while state |1> describes the condition where the conduction band state is vacant and the valence band state is occupied [128]. To describe such a system, a basic model can be constructed in which we suppose each emitter to interact with all modes of the cavity but ignore direct interaction among emitters ([129], §9 and [130]). The cavity modes, on their part, undergo damping as a result of loss at the cavity boundaries, and the loss can be modeled as a thermal reservoir. Loss at the cavity boundary, such as loss in a mirror, or loss of energy through the mirror and its eventual conversion to heat at some remote point in space, generally satisfies the assumptions of a reservoir model: it is weak interaction with a large stochastic system that is disordered and does not retain memory of past interactions. Further, this reservoir is passive, as it does not return energy to the mode. Rather, it drains the mode energy over time and is commonly known as the zero temperature condition. The Hamiltonian describing each single emitter in this system can be expressed as ^ ¼H ^A þH ^F þH ^ AF þ H ^R þH ^ FR H

ð3:7Þ

^ A, H ^ F , and H ^ R are the emitter, field, and reservoir Hamiltonian, respectively. where H ^ ^ FR denotes H AF denotes interaction between the emitter and the field modes, while H interaction between the field modes and the reservoir. We note that even if, by assumption, a given emitter does not directly interact with other emitters, the field modes still interact with all emitters present, rather than only with a single emitter. This interaction is not included in the Hamiltonian in Equation (3.7), either explicitly or as part of the reservoir. Here, we adopt the simplified model as a starting point to illustrate how it leads to the expressions for the Purcell factor commonly found in the literature [7, 35, 124, 125, 131, 132]. We discuss in Section 3.3 that the effect of the emitter population on the field modes cannot justifiably be ignored in semiconductor lasers. In a system where an emitter interacts with the field, and the field interacts with a thermal reservoir, the results summarized in Appendices A.1 and A.2 apply directly. The cavity Purcell factor Fp is defined as the ratio of spontaneous emission in a cavity to that in free space. In the evaluation of Fp, it is common to replace the vacuum free space emission probability presented in Equation (A.10) in Appendix A by the emission

70

Purcell Effect and the Evaluation of Purcell and Spontaneous Emission Factors

probability of bulk material of effective index nr, with no cavity [7, 124]. The spontamaterial , takes the same form as in free neous emission probability in the bulk material, P2→1;j0...0〉 space, except that ε0 is replaced by the permittivity of the medium εr ¼ n2r ε0 and c is scaled down by the refractive index nr. It is expressed as (based on Equation (A.10) in Appendix A) ð material P2→1;j0...0〉

≈ ≈

ω321 3πℏεr ðc=nr Þ3 ω 321

3πℏεr ðc=nr Þ

τcoll j ℘12 ðω21 Þj2 Dðω21 Þdω21 ð3:8Þ

τ j ℘12 ðω 21 Þj2 3 coll

where ω21 is the mode resonant frequency, ℘12 ðω21 Þ is the dipole matrix element, and D(ω21) characterizes the inhomogeneity of the system. The intraband collision time, τcoll, is the average time between carrier-carrier and carrier-phonon collisions, and it decreases with increasing temperature [133]. In the second line of Equation (3.8), ω321 and ℘12 ðω21 Þ are evaluated at the center frequency ω 21 of the inhomogeneous broadening spectrum D(ω21) and are pulled out of the integration, because these quantities vary relatively little over the homogeneous broadening range. In a damped cavity, the mode interacts with the reservoir. Provided that equilibrium between the mode and the reservoir is reached, we can obtain the photon emission probability in steady-state, Xω  ð k cav P2→1;equilibrium ¼ n ðωk Þ þ 1 j ℘12 ðω21 Þ  ek ðre Þj2 Dðω21 Þ ℏ k ð ð3:9Þ  Lk ðω  ωk ÞRðω  ω21 ; τcoll Þdωdω21 where R(ω−ω21,τcoll) is the homogeneous broadening function and depends on τcoll. ð a function of ω, R(ω) peaks at ω21, has a width on the order of 1/τcoll, and Viewed as satisfies Rðω  ω21 ; τcoll Þdω ¼ 2π  τcoll [134]. The Lorentzian Lk(ω−ωk) in Equation (3.9) is expressed as 2 1 Δωk 2 1 2 Q Lk ðω  ωk Þ ≡

¼ 

; 2 2 π 1 π ωk 1 2 2 Ck þ ðω  ωk Þ Δωk þ ðω  ωk Þ 2 2 where Ck ¼ Δωk 1 Ck 2



ð3:10Þ and the quality factor is defined as Q ≡ ωk =Δωk , where Δωk is the FWHM of cavity lineshape. The convolution in Equation (3.9) determines the emission probability in a

3.2 Formulation of Purcell Effect in Semiconductor Nanolasers at Room Temperature

71

cavity for an inhomogeneously broadened ensemble of emitters, when the mode-reservoir equilibrium has been reached. The effect of the reservoir on the emission probability is described by Lk(ω−ωk), whose spectral property is described by Equation (3.10). The Purcell factor of the cavity mode is denoted as FP,mnp, indexed by the m, n, p mode in Cartesian coordinates: FP;mnp ≡

cav P2→1;equilibrium

material P2→1;j0...0〉 ð X 3πεr ðc=nr Þ3 ωk 1 j ℘12 ðω21 Þ  ek ðre Þj2 Dðω21 Þ ¼ 3 2 τ ω coll ð ω Þj j ℘ 21 21 12 k

ð  Lk ðω  ωk ÞRðω  ω21 ; τcoll Þdωdω21 ≈

X 3πεr ðc=nr Þ3 ωk j ℘ ðω 21 Þ  ek ðre Þj2 12 3 τ ω coll j ℘12 ðω 21 Þj2 21 k ¼ mnp ð ð  Dðω21 Þ Lk ðω  ωk ÞRðω  ω21 ; τcoll Þdωdω21

ð3:11Þ

The emission probability in Equation (3.9), and hence the Purcell factor in Equation (3.11), depends on the location re of the emitter. More precisely, it depends on the normalized mode field at the location of the emitter ek ðre Þ, as well as on the orientation of the emitter’s dipole moment matrix element ℘12 ðω 21 Þ relative to the field. If the emitters are randomly oriented and uniformly distributed over an active region of volume Va, the quantity j ℘12 ðω 21 Þ  ek ðre Þj2 is replaced by its average overall locations and orientations: ð 1 1 j ℘12 ðω 21 Þ  ek ðre Þj2 → j ℘12 ðω 21 Þj2 jek ðrÞj2 d 3 r ð3:12Þ 3 Va Va where the coefficient 1/3 accounts for the random emitter orientation. In certain situations, the carrier distribution over Va may become nonuniform. For example, in MQW structures, the carrier distributions in the well and barrier regions differ significantly. Even in bulk semiconductors, the recombination of carriers may vary spatially, with the highest rates occurring at field antinodes. This is the case if the recombination at field antinodes is so rapid that diffusion of carriers from other parts of the active volume is not fast enough to avoid depletion. Carrier depletion at field antinodes and subsequent diffusion from the nodes toward the antinodes leads to the spatial inhomogeneity of the recombination. At room temperature, the diffusion length of carriers in InGaAsP (i.e., average distance traveled before recombination) is on the order of 1–2 μm [135]. The distance between the field node and antinode in visible and near infrared subwavelength semiconductor cavities, on the other hand, is usually less than 0.5 μm [34, 35]. Thus, the depletion regions would remain relatively depleted due to the finite diffusion time. Under these circumstances, Equation (3.12) should then be replaced by an appropriately weighted average:

72

Purcell Effect and the Evaluation of Purcell and Spontaneous Emission Factors

8 > > ð > > > > > ε jEk ðrÞj2 d 3 r r X πðc=nr Þ3 ωk 1 <   V a FP ¼ 1 3 3 V >ð 2 0 0 0 τ ω coll a > 21 ∂ ω ε ð r; ω Þ k R > > > 4@ þ εR ðr; ωk ÞAE2k ðrÞ5d 3 r > > : 0 ∂ω0 ð 

ð

ω ¼ωk

g

Dðω21 Þ Lk ðω  ωk ÞRðω  ω21 ; τcoll Þdωdω21

ð ð X πðc=nr Þ3 ωk 1 ¼ fΓ g D ð ω Þ Lk ðω  ωk ÞRðω  ω21 ; τcoll Þdωdω21 k 21 τcoll ω 321 Va k X ¼ FP;mnp ð3:13Þ k ¼ mnp

where Γk is the energy confinement factor of mode k. Equation (3.13) permits several observations. First, the double integral in Equation (3.13) is the convolution of inhomogeneous broadening D(ω21), cavity Lorentzian Lk(ω−ωk), and homogeneous broadening R(ω−ω21,τcoll). It should be noted that although the homogeneous broadening function R(ω) and the inhomogeneous broadening function D(ω) appear symmetrically in Equation (3.13), they may in principle exhibit different dynamics. In particular, rapid recombination of carriers near the mode frequency ωk may deplete the carrier population at that frequency faster than it is replenished by intraband scattering (this phenomenon is known as “spectral hole burning”). In such cases, it could be meaningful to disaggregate the integral in dω21 in Equation (3.13) and define separate Purcell factors for carriers at different frequencies ω21 [136]. More typically, however, especially at room temperatures, III-V semiconductor materials have sub-picosecond intraband relaxation time τcoll (e.g., τcoll ~0.3 ps for InGaAsP), which is much shorter than photon emission time. Under this circumstance, the distribution of carriers D(ω21) is at all times the equilibrium distribution ([137], appendix 6). This equilibrium distribution closely resembles the photoluminescence spectrum [138]. In semiconductor lasers utilizing bulk or MQW gain material, it is the broadest of the three convolution factors in Equation (3.13) and therefore dominates the convolution. For InGaAsP at room temperature, the FWHM of D (ω21) and R(ω−ω21,τcoll) are approximately 7∙1013 rad/s and 6.7⋅1012 rad/s, respectively. D(ω21) dominates the convolution in Equation (3.13) as long as the cavity Q factor is above 19, which corresponds to a FWHM of 7⋅1013 rad/s. For practical cavities, the Q factor will be significantly larger, thus diminishing the contribution of Lk(ω−ωk) to the resulting Purcell factor. In fact, R(ω−ω21,τcoll), alone, dominates Lk (ω−ωk) if the Q factor is greater than 200 [133, 139]. Consequently, in typical III-V semiconductor lasers with MQW or bulk gain material, the cavity Q factor plays a negligible role in determining the spontaneous emission rate and FP. Further, while the cavity lineshape broadens with temperature for well-confined cavity modes, the homogeneous lineshape broadens as well.

3.3 Applicability of the Formulation

73

Second, FP may be large in small laser cavities due to its inverse proportionality to the active region volume Va. However, FP is actually inversely proportional to the effective size of the mode, Va =Gk , where the mode-gain overlap factor Γk is defined in Equation (3.13) and describes the spatial overlap between the mode and the active region. Thus, if the mode is poorly confined, Γk 0), are associated with a discrete spectrum of eigenvalues. In contrast, the radiative continuum of modes (denoted by R in Figure 9.3) are described by modes with a purely oscillatory behavior (Re[γ1], Re[γN] = 0). The third class of modes, socalled leaky modes (denoted by L in Figure 9.3), show a divergent behavior in the cladding. These modes, despite not being part of the complete basis set, are observable as resonances in the reflection spectrum. A specific, yet characteristic example structure consisting of a planar metal-dielectric stack, which is to operate at near-infrared (telecommunication) wavelengths, is designed

206

Cavity-free Nanolaser

L

B

R

Air

ITO

d1

Si

d2

y x

ITO

z

Figure 9.3

Schematic illustration of the TM2 leaky (L), bound (B), and radiation (R) modes of the metaldielectric-metal heterostructure. Reprinted from reference [260] with permission from American Physical Society (APS).

Figure 9.4

Geometry and dispersion of the planar stoplight structure in the passive case. (a) The metaldielectric structure with gain section thickness h = 290 nm and top metal layer thickness t = 500 nm is designed to support a weakly leaky TM2 mode at a wavelength of 1547 nm. (b) Passive case: The (complex frequency) dispersion of the TM2 mode exhibits two stoplight points (marked with solid circles) within the light cone (dashed line) that are lined up with the gain spectrum centered around 193 THz. Reprinted from reference [74] with permission from Macmillan Publishers Ltd.

by Pickering et al. [74]. It consists of a top ITO layer that acts as metal but at the same time allows light extraction at a thickness of 500 nm. Depending on whether it is a passive waveguide design or an active laser design, the center dielectric can be either silicon (for waveguide design) or a combination of silicon and InGaAsP (for laser design) with a height of 290 nm, and the bottom layer is a thick ITO substrate. This structure resembles the five-layer stack in Figure 9.2(c) with different metal thicknesses. Note that in the laser design, although the dielectric region is divided into active InGaAsP and passive Si sections, no cavity is constructed in the conventional sense because InGaAsP and Si both have refractive indices of 3.4 around 1.55 μm. For now, it suffices to restrict to the passive dispersion relation analysis by taking the entire center dielectric region to be either lossless InGaAsP or Si (Figure 9.4(a)). Finally, a 10 nm Si buffer is added atop the metal. The buffer layer is used to reduce gain quenching in close vicinity to the metal.

9.2 Effect of Surface Roughness on Light Stopping

207

The choice of structure and materials gives rise to a (weakly) leaky TM2 mode in the dispersion diagram (Figure 9.4(b)), which, by design, features two stoplight points that lie close in frequency: ω1/2π = 193.8 THz (λ1 = 1546.9 nm) and k1 = 0 and ω2/2π = 193.78 THz (λ2 = 1547.06 nm) and k2 = 1.42 μm−1. With both stoplight points situated inside the light cone, the mode is therein weakly leaky. For the (weakly leaky) structure under investigation, leaky modes accurately capture the changes to the modal dispersion and the additional radiative loss caused by the coupling of the waveguide mode to the continuum of free-space modes. It is expected to couple to free-space radiation perpendicular to the plane, as schematically illustrated in Figure 9.4(a). The cavity Q factor and the collectable amount of emission can then be controlled by adjusting the thickness of the top metal-cladding. With the top ITO thickness being 290 nm, around 3% to 4% of generated light can be collected.

9.2

Effect of Surface Roughness on Light Stopping For optical devices utilizing slow/stoplight phenomena, stringent requirements on material surface and interface smoothness are a major hurdle in their practical realization. For example, even small structural imperfections can have a detrimental impact on the propagation characteristics at stoplight points in photonic crystal-based slow- or stoplight devices. This is because in photonic crystal-based devices, the slowing down of light is achieved via periodic back reflection by a lattice of scatters. These scatters have sizes comparable to the wavelength of light that is propagating within [261]. Nanometer scale interface roughness, therefore, destroys the theoretically assumed perfectly periodic structure, and the increased light scattering at rough surfaces leads to a “smearing out” effect in the attained group indices at band edges [262]. Here, it is structural disorder that fundamentally limits the attainment of light stopping. In the stoplight waveguide and laser design, the stoplight behavior arises from opposing energy flows inside the metallic cladding rather than backscattering at geometric features such as in the case of photonic crystals. Time-resolved simulation of the energy flow of a passive structure can be used to analyze the effect of surface roughness on the localization of energy [237]. Figure 9.5(a) depicts the passive structure with a rough surface. Surface roughness, whose magnitude follows a normal distribution, is introduced both at the Si/ ITO as well as the ITO/air interface. Tsakmakidis et al. investigated three realistic roughness values: root-mean-square (r.m.s.) of 0.5 nm, 1 nm, and 3 nm [260]. At a critical incidence angle of θSL ≈ 17.6° from a Gaussian spatial shape pulse, light can be stopped in the waveguide in the absence of surface roughness. Incident light at angles ±4° away from θSL are also studied. The right columns of Figure 9.5(b)–(e) show the temporal variations of the mean energy position of the exited pulse. With a perfectly smooth surface, the pulse is immobilized at = 0 as shown in Figure 9.5(b). In the case of both smooth surface (Figure 9.5(b)) and r.m.s. = 0.5 nm roughness (Figure 9.5(c)), the pulse moves forward or backward, for angles that are +4° or −4° different from θSL, respectively. The cluster of lines around each incident angle corresponds to different samples. As the surface roughness increases to 1 nm, the pulse moves backward for all angles, as all the propagation lines in Figure 9.5(d) are below = 0. We see from Figure 9.5(b)–(d) that for

208

Cavity-free Nanolaser

(a)

+4º θSL –4º

t h y x

(b)

15

RMS = 0nm

Hz

10

+4º

0

5 17.6º

–1 F

0

Sx

–5

–4º

(nm)

+1

–10

B

+1

15

Hz

RMS = 0.5nm

0 –1 F

10 +4º

5 0

Sx

–4º

–5

(nm)

(c)

–15

–10

B

+1

15 Hz

RMS = 1nm

10

0 –1 F

5 +4º Sx –4º

0 –5

(nm)

(d)

–15

–10

B

RMS = 3nm

Hz

10

0 –1 F

15 5 0

Sx

–5

(nm)

(e) +1

–15

–10

B 80

Figure 9.5

100 120 x position μm

140 1.05 1.08 1.11 t (ps)

–15 1.14

(a) The slow-light mode is excited by an electromagnetic pulse of Gaussian spatial shape incident under an angle θ. The critical incidence angle for stopping light is θSL ≈ 17.6°, under which the SL

9.3 Design of Stoplight Nanolasers

209

r.m.s. roughness less than 1 nm, the energy velocity remains constant in time with a clear dependence on the injection angle. It is important to note that for these roughness values, one can always find an optimum excitation angle for which the energy velocity along the waveguide direction is zero. The optimal angle depends on the level of surface roughness and, additionally, varies from sample to sample. In contrast, when the r.m.s. roughness is increased to 3 nm (Figure 9.5(e)), the nature of pulse propagation changes dramatically: the energy velocity is no longer constant and the pulse is equally likely to propagate in a forward or backward direction. In this regime, the propagation of energy is diffusive and correlates only weakly with the waveguide dispersion. As a result of the strong scattering at the surface inhomogeneities, one can observe the pulse breakup, visibly shown in the x-component of the Poynting vector Sx (left panel), and the complete stopping of light is no longer possible (right panel). Next, we incorporate the InGaAsP gain material into the center dielectric region. Different from the passive structure analysis in which an external pulse is launched on the structure, here the loss of energy that is generated within the gain medium can be studied. The amount of loss due to the rough surface presents another evaluation category. Figure 9.6 shows that up to r.m.s = 1.5 nm roughness, the laser is resilient to surface roughness and total loss rate is minimal. Above 1.5 nm roughness, however, the average loss rate steadily rises from the scattering of the modal fields at rough interfaces and the increased absorption of energy at hotspots. The insets of Figure 9.6 show the mode profiles at different levels of surface roughness. For r.m.s. roughness above 2.5 nm, an increasing number of surface configurations do not show the existence of the stoplight mode, in agreement with the passive structure analysis.

9.3

Design of Stoplight Nanolasers So far, we have focused on the analysis of either lossless active structures or passive structures, both in the ideal smooth material surface and interface case in which an extended band of flat dispersion can be expected (Figure 9.4(b)), and in the practical non-smooth interface case in which r.m.s. roughness of 3 nm destroys the flat dispersion (Figure 9.5(e)).

Caption for Figure 9.5

(cont.)

point can be excited in the absence of surface roughness. (b–e) Left column: two-dimensional (xz-plane) spatial distribution of the magnetic field component (Hz) and Poynting flux (Sx) of the excited light pulse for different r.m.s. amplitudes of surface roughness (r.m.s.= 0–3 nm). Right column: temporal variations of the mean energy position of the excited pulse, inside the waveguide, for illumination angles of the exciting light beam between +4° and −4°, compared with the “critical” angle θSL. (a) is reprinted from reference [74] with permission from Macmillan Publishers Ltd; (b)−(e) is reprinted from reference [260] with permission from American Physical Society (APS).

210

Cavity-free Nanolaser

Table 9.1 Gain medium parameters for emission into TM2 mode. Symbol

Description

Value

ωe/2π(λe) γe σ0,e N ge

Emission frequency (wavelength) Emission spectral half-width Emission cross section Carrier density Maximum gain coefficient

193.41 THz (1550 nm) 1/(40fs) = 25 ps−1 (≈ 32 nm) 2.09∙10−15 cm2 2∙1018 cm−3 4180 cm−1

Reprinted from reference [74] with permission from Macmillan Publishers Ltd.

Figure 9.6

Impact of surface roughness on stoplight lasing. For small values of surface roughness, the total loss rate of the lasing mode is weakly dependent on the r.m.s. surface roughness. Above r.m.s. values of 1.5 nm, a linear increase is observed. Surface roughness leads to the appearance of local hotspots in the time-averaged intensity profile during steady-state lasing, shown here for increasing levels of surface roughness: 1, 2, and 2.5 nm from bottom to top. The total loss rate and its error were calculated as the average loss rate and s.d. from 10 lasing configurations for each r.m. s. surface roughness value. Reprinted from reference [74] with permission from Macmillan Publishers Ltd.

Next, the threshold condition in an active structure of the same geometry can be evaluated, with the help of small-signal gain analysis. The analytic dispersion equations can be solved for various levels of inversion (see Table 9.1 for the gain material parameters). Figure 9.7 plots the analytical solutions. As the two stoplight points retain their positions and effectively pin down the dispersion between them, no significant change of the modal dispersion is observed with increasing inversion (Figure 9.7, top). Thanks to the flatness of the dispersion band, an average group velocity of ≈ 3×10−4 c is obtained for all inversion levels (Figure 9.7, center). In contrast, the modal loss displays a strong

9.3 Design of Stoplight Nanolasers

Figure 9.7

211

Small-signal gain analysis of the planar stoplight structure. When gain is introduced into the core, the modal dispersion, group velocity, and modal loss (top to bottom) will change depending on the gain coefficient ranging from cm−1 (no inversion) to 4180 cm−1 (maximum inversion). The analysis highlights the robustness of the SL points, leading to a flat dispersion of average velocity ≈ 3×10−4 c that is practically not affected by the level of inversion. The dashed line in the top panel represents the light line. Reprinted from reference [74] with permission from Macmillan Publishers Ltd.

dependence on the inversion and changes sign at ΔNth ≈ 0.13 (Figure 9.7, bottom). At this point, the mode becomes amplified and, owing to the flatness of the band, experiences uniform gain over a broad range of k-values (up to k ≈ 5 μm−1 where the dispersion ω(k) moves out of the spectral window of gain). As the calculated loss rate includes both dissipative and radiative (leakage through the top ITO layer) contributions, one can associate ΔNth with the threshold inversion required for lasing. To obtain insight into the transient dynamics and lasing characteristics enabled by the cavity-free nanolaser design, light-light curves and emission linewidth behavior can be obtained using time-resolved rate equation simulations. Recall that in the laser design, the center dielectric is composed of a finite width InGaAsP surrounded by Si. Although the waveguide is divided into active (InGaAsP) and passive (Si) sections, because InGaAsP and Si both have refractive indices of 3.4 around 1.55 μm, no cavity is constructed in the conventional sense. Initially, the gain section will be a strip of 400 nm width, approximately four times smaller than the vacuum wavelength, but other gain widths can be explored as well. Figure 9.8 shows the steady-state light-light curve in log scale, which follows the familiar S-shape with a threshold pump rate of

212

Cavity-free Nanolaser

Figure 9.8

The laser output energy shows an S-shaped input-output curve with a threshold pump rate of rp ≈ 1.43 ps−1. A comparison of the amplified spontaneous emission (ASE) spectrum below threshold (left inset) and the lasing spectrum (right inset) indicates spectral narrowing due to the buildup of phase coherence in the SL mode. Close to its maximum value, the ASE spectrum closely follows the spectral density of states of the TM2 mode at k = 0 (narrow dashed curve, normalized to the ASE and lasing peak values, respectively) and is much narrower than the gain spectrum (broad dashed curve). The output energy and its error in c were determined during steady-state emission as the average value and s.d. of the cycle-averaged electromagnetic energy in a fixed time window. Reprinted from reference [74] with permission from Macmillan Publishers Ltd.

rp ≈ 1.43 ps−1. The left inset of Figure 9.8 shows the emission line below threshold, together with the gain spectrum (broad dashed line) and the spectral density of states of the mode at k = 0 (narrow dashed line). The emission line represents amplified spontaneous emission, whose spectrum follows the shape of the spectral density of states at k = 0. As pumping increases, the emission line narrows as a result of the buildup of phase coherence in the stoplight mode (right inset of Figure 9.8). The gain region can be optically pumped using a propagating waveguide mode at a higher frequency, and the un-pumped Si regions of the waveguide will not contribute loss to the system. The pump wavelength not only needs to be shorter than that of the stoplight mode but also needs to reside on the dispersion curve where propagating modes are supported. High-k modes of the TM2 branch (Figure 9.4(b)), for example, satisfy both requirements and can therefore be utilized as a pump. In this scheme, end-fire coupling of the pump through one end of the waveguide is possible. Figure 9.9(a) presents the injection of the pump field from the negative x-direction (also schematically shown in Figure 9.1(a)). It has the TM2 mode profile in the Si regions, represented by the two non-zero components Ex (top) and Ey (bottom). The pump field’s interference with the stoplight lasing mode leads to a strong deviation from the pump mode profile, as observed close to and inside the gain section, particularly for the Ex (top) component. Figure 9.9(b) shows the lasing mode intensity profile and the Poynting vectors of the lasing mode on spectral filtering of Figure 9.9(a) around the lasing frequency, and the normalized carrier inversion in steady state is shown in Figure 9.9(c). Other optical

9.3 Design of Stoplight Nanolasers

Figure 9.9

213

(a) Snapshots of the electric field amplitudes Ex and Ey showing the incoming pump field (from the left) and, close to and inside the gain section, its interference with the SL lasing mode. Approximately 19% of the pump mode energy is absorbed in the 400-nm-wide gain section. (b) The intensity profile and Poynting vectors of the steady-state lasing mode and (c) the associated steadystate inversion. Reprinted from reference [74] with permission from Macmillan Publishers Ltd.

pumping schemes are possible too, for example, through metal grating on top of the top ITO layer at an angle. With the planar structure analyzed so far being isotropic in two dimensions, the generalization to three dimensions is straightforward. By introducing a small cylinder of gain material (or, e.g., a quantum dot) into the waveguide core layer or by spatially selective excitation of a homogeneous gain layer, one can conceivably construct a photonic-dot laser that, evanescently pumped by a near-field tip, would receive feedback by formation of a three-dimensional stoplight vortex. In the vicinity of the metal layer, high gain coupling factors can be expected due to the combined action of a vertical plasmonic confinement and the in-plane stopping of light. In principle, the absence of a cavity enables simpler fabrication compared to other plasmonic laser architectures. Additionally, use of the slow-light effect suggests that the threshold gain may be reduced relative to other plasmonic sources. However, the requirement of the stoplight nanolasers on surface smoothness remains a challenge. Experimental demonstrating of the cavity-free stoplight nanolaser remains an open problem.

10 Beyond Nanolasers: Inversionless Exciton-polariton Microlaser

Population inversion of the gain medium is a fundamental prerequisite for lasing, according to most textbooks. Exceptions to this prerequisite, however, have been shown to exist. For example, lasing without inversion, which relies on an electromagnetic-induced transparency phenomenon to enable lasing, was demonstrated in atomic [129] and semiconductor systems [263]. Research over the past two decades has also shown a different inversionless lasing mechanism: exciton-polariton condensation, which can be realized in optical microcavities. Polariton lasing represents a fundamentally different and potentially more efficient process for generating coherent light than population inversion [264]. As such, it is promising for ultralow threshold, highly energy-efficient sources of coherent light. From earlier demonstrations, polariton lasers were deemed impractical because of the implementation of magneto-optical setup of magnetic fields of up to several Tesla, in addition to their cryogenic temperature operation, which assisted the condensation of exciton-polaritons needed for lasing. However, in 2014, a room-temperature, electrically pumped semiconductor polariton laser without applied magnetic field was demonstrated for the first time, marking a major milestone in the application of polariton lasers. The device structure of a polariton laser is typically analogous to that of a conventional micro-sized vertical-cavity surface-emitting laser (VCSEL), but it is not hard to envision the device becoming smaller in size, as the field matures [12, 51]. From a fundamental point of view, the operation of polariton lasers relies on the establishment of an entity of highly correlated degenerate exciton-polaritons, a solidstate Bose-Einstein condensate (BEC). Because the BEC is achieved with excitonpolaritons, it may be established at much higher temperatures than atomic BECs [265]. Thus, in addition to their promise as highly efficient sources of coherent radiation, polariton lasers offer an economical and easily attainable environment for the study of macroscopic quantum phenomenon and can help move the field of light-matter coupling into a more applied arena. For these reasons, we dedicate a chapter in this book to microscale, ultralow threshold, inversionless polariton lasers.

10.1

Background Exciton-polaritons are quasi-particles formed in resonators that provide strong coupling between intracavity photons and the excitonic states of a gain medium (Figure 10.1).

10.1 Background

Figure 10.1

215

Illustration of VCSEL structure used to study exciton–polariton condensation. The planar Bragg mirrors enclosing a quantum-well–based gain medium quantize the longitudinal wavevector (kz) without restricting the in-plane wavevector (k||). Reprinted from reference [265] with permission from Macmillan Publishers Ltd.

In other words, a system is best described in a coupled exciton-polariton picture − rather than by separately considering photons and excitons − if the rate at which photons escape from the cavity is smaller than the rate at which excitons are converted into photons and vice versa (i.e., the rate of stimulated emission and absorption). This condition is also known as the “strong coupling.” In this strong-coupling regime, exciton-polaritons can form a condensate or coherent state once their density becomes sufficiently high. Photon leakage from a resonator containing such a condensate yields coherent light that is nearly indistinguishable from conventional laser emission − hence the term “exciton-polariton laser” (sometimes abbreviated as “polariton laser”). Strong coupling can be achieved, for instance in small Fabry-Perot cavities, by locating the gain medium at the antinodes of the cavity photon mode (i.e., VCSEL-type structures). Exciton–polariton condensates exhibit unique and rich physics, and their fundamental properties have been studied intensively [266]. The device structure of a polariton laser can be analogous to that of a conventional VCSEL, and polariton lasers show conventional photonic lasing if excited above population inversion. In fact, the presence of two distinct thresholds, the lower associated with polariton lasing and the higher associated with conventional photon lasing, is regarded as a sign of polariton condensation [51, 264]. As polariton condensation does not require inversion, such lasers have the potential to outperform conventional semiconductor lasers through operation via the strong-coupling regime as opposed to their weak-coupling counterparts in that they have a lower threshold density. Although this threshold can be very small, one does not expect unity-β factor lasing behavior (e.g., that analyzed in Chapter 4, Sections 4.5 and 4.7). To understand the working principle of the polariton laser, we first briefly review this field [266].1 Two well-known phenomena arising directly from Bose statistics are (i) 1

For more reading on semiconductor cavity QED in the strong-coupling regime, refer Semiconductor Cavity Quantum Electrodynamics [266].

216

Beyond Nanolasers: Inversionless Exciton-polariton Microlaser

ω

UP

LE

ωL ωT

LP 2ωc

k0 Figure 10.2

k

Schematic representation of the dispersion of the upper and lower polaritons and of the longitudinal exciton (solid lines). The dispersion of the uncoupled photon and transverse exciton modes is also shown (dashed lines). Reprinted from reference [266] with permission from Springer Publishing.

BEC of massive bosonic particles in thermal equilibrium (most notably for 87Rb atom), featuring the bosonic behavior of atoms and excitons and (ii) photon lasers: coherent light generation from incoherent nonequilibrium inverted reservoirs, featuring the bosonic behavior of photons. Even though both topics have been extensively studied, little was known about their connection until the proposal of the polariton laser by Imamoglu et al. [50]. Polaritons are the normal modes of optically active excitons and a radiation field in solids. In other words, the polariton involves a spatially coherent coupling of the exciton with the optical field. In an infinite crystal, only those excitons at the crossing of the noninteracting exciton and photon dispersions can decay, owing to the requirement of energy-momentum conservation in the radiation process (Figure 10.2). However, the exciton-photon coupling leads to an anti-crossing in the polariton dispersions. In Figure 10.2, the resulting two normal modes of lower and higher eigenenergies from the coupling are termed upper polariton (UP) and lower polariton (LP), respectively. Thus, a polariton is a linear superposition of an exciton and a photon with the same in-plane wave-number k||. Since both excitons and photons are bosons, so are the polaritons. Figure 10.3 summarizes the polariton laser: it can be understood in contrast to both a BEC in thermal equilibrium and the photon laser. For a thermal equilibrium reservoir and a vanishing photon character of the polaritons, one obtains a BEC of excitons. In the opposite limit of a nonequilibrium inverted reservoir and a vanishing exciton character, the polariton laser is indistinguishable from a photon laser. We will refer to Figure 10.3 again in later sections.

10.2 Strong Coupling and Condensation between Quantum-well Excitons and Cavity Photons

217

Figure 10.3

Schematic of differences between the excitonic BEC, exciton-polariton laser, and photon laser. Reprinted from reference [50] with permission from American Physical Society (APS).

Figure 10.4

The operational principles of (a) polariton BEC and (b) photon laser. Reprinted from reference [264] with permission from American Physical Society (APS).

Figure 10.4 graphically illustrates the difference in the operation principle between polariton and photon lasers: polariton lasers rely on stimulated phonon and polariton-polariton scattering, while photon lasers rely on stimulated photon emission.

10.2

Strong Coupling and Condensation between Quantum-well Excitons and Cavity Photons The strong coupling regime is a requisite for the generation of coherent radiation via the stimulated scattering of polaritons. Whether the coupling between two oscillators is

218

Beyond Nanolasers: Inversionless Exciton-polariton Microlaser

strong or weak depends on the magnitude of the coupling constant relative to the natural decay rates of the two oscillators. In the context of a semiconductor, for example, multiple-quantum-wells (MQWs) (the gain material used in experimental demonstration of semiconductor polariton lasers [12, 51]), the two oscillators are the dipole formed by a bound electron-hole pair in the semiconductor, that is, the exciton and the dipole formed by the photonic mode of the cavity. The coupling constant between an exciton and a cavity photon may be expressed as [266] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð ð ð     f Ω¼q Fk∥ ;n r∥ ; z vk 0∥ ;n0 r∥ ; z d 2 r∥ dz 8εLw me S

ð10:1Þ

V

where q, f, ε, Lw, me, S are the elementary charge, oscillator strength, electrical permittivity, quantum-well thickness, free electron mass, and sample area, respectively. The oscillator strength of the exciton dipole f of Equation (10.1) is a material property and is defined by f ¼

2mω0 2 x ℏ 12

ð10:2Þ

where m, ω0, χ12 are effective mass, oscillation (transition) frequency, and dipole length, respectively. Note that the oscillator strength is also a monotonic function of an external magnetic field. The first function inside the integral of Equation (10.1) is the center-of-mass wavefunction of the two-dimensional exciton confined in the QW, Fk∥ ;n



 r∥ ; z ¼

rffiffiffiffiffiffi

 2 πnð z  z0 Þ eik∥ ∙r∥ pffiffiffi sin Lw Lw S

ð10:3Þ

The second function within the integral of Equation (10.1) is the wavefunction of a longitudinal cavity photon mode,   vk 0∥ ;n0 r∥ ; z ¼

rffiffiffiffiffiffi 0  ik 0 ∙r 2 πn z e ∥ ∥ pffiffiffi sin Lw Leff S

ð10:4Þ

where Leff is the effective length of the cavity. On inspection of Equation (10.1), we see that the coupling constant is just the overlap integral of the exciton and cavity photon wavefunctions, scaled by the square root of the oscillator strength. All else being equal, the coupling constant reaches a maximum when an antinode of the cavity mode is located in the center of the quantum well, while the coupling constant vanishes when a node of the cavity mode is aligned with the quantum well. In Equations (10.1)−(10.4), we have assumed that longitudinal direction of the cavity and the growth direction of the QWs in the z direction, while the exciton is unconfined in the xy-plane (Figure 10.1). The dimensions of both wavefunctions are Length−3/2 and the unit of the coupling constant is s−1; that is, it is a rate. The strong-coupling regime occurs when the coupling rate

10.2 Strong Coupling and Condensation between Quantum-well Excitons and Cavity Photons

219

significantly exceeds the decay rates of the quantum-well exciton and the cavity photon, ζex, ζcp. The weak-coupling regime, on the other hand, occurs when the inequality is reversed [266]: Ω≫ζ ex ; ζ cp Ω≪ζ ex ; ζ cp

strong coupling; weak coupling

ð10:5Þ

In a microcavity, the quality factor characterizes the amount of energy stored relative to that dissipated. From a more experimental viewpoint, the quality factor may also be expressed as the ratio of the resonant frequency to its spectral fullwidth-at-half-maximum (FWHM). It is related to the cavity photon decay rate through [137]: Q≜

ωcp Δωcp

Qω0 ¼ τcp ¼ ζ 1 cp

ð10:6Þ

where ωcp, Δωcp, τcp are the resonant frequency, FWHM, and lifetime of the cavity photon, respectively. For most practical structures, the quality factor is extracted from numerical simulations (e.g., through a commercial Finite Element Method (FEM) package). The quality factor may be expressed in terms of the real and imaginary parts of the eigenfrequency of a particular mode: Qk ¼ 0

Im½λcp;k0  2  Re½λcp;k 0 

ð10:7Þ

0

In Equation (10.7), k denotes the k th eigenvalue of the infinite, discrete set of solutions to the cavity problem being modeled. The real and imaginary parts of the eigenfrequency of Equation (10.7) are associated with the decay rate and energy of the 0 k th cavity photon mode, respectively. To obtain quality factors high enough for strong coupling, distributed Bragg reflector (DBR) in VCSEL cavities are often used [12, 51]. These cavities can have extremely high reflectivity, increasing monotonically with the number of periods of the Bragg reflectors. The energy and decay rate of the exciton may be determined using the transfer matrix method, with the displacement field in the QW adjusted to account for the exciton. According to reference [266], the electrical susceptibility that arises from excitonconfined states in a QW may be written as a nonlocal function: χðω; k; z; z0 Þ ¼ with

gs j〈uc jqrjuv 〉j2   ρðzÞρðz0 Þ ℏωex k∥  ω  jζ ex

  ρð zÞ ¼ Fk∥ ;n r∥ ; z ¼ 0 ce ðzÞvh ð zÞ

ð10:8Þ

ð10:9Þ

220

Beyond Nanolasers: Inversionless Exciton-polariton Microlaser

In Equation (10.8), gs accounts for spin-orbit interaction and is on the order of unity. Equation (10.9) represents the product of the exciton envelope function at zero electronhole separation with the confinement functions of the electrons and holes in the well. While the displacement field in regions outside the QW is simply DðzÞ ¼ n2 ε0 Eð zÞ, the displacement field within the QW, due to the nonlocal susceptibility of Equation (10.8), is [266] ð 2 Dð zÞ ¼ n∞ EðzÞ þ 4π χðω; k; z; z0 ÞEðz0 Þdz0 ð10:10Þ With the constitutive relation, Equation (10.10), established, the transfer matrix method proceeds as usual to obtain the reflection and transmission coefficients as functions of wavelength and incident angle. The energy and decay rates for a given location on the plane correspond to the real and imaginary parts of the poles of the reflection and transmission coefficients, ξex,k, i.e., ωex;k ¼ Re½ξ ex;k  and ζ ex;k ¼ Im½ξ ex;k . With the coupling constant, exciton decay, and cavity photon decay rates all known, the operational regime may be determined by evaluating Equation (10.5). In general, the decay rate of the cavity photon exceeds that of the QW exciton [267]. Experimentally, Weisbuch et al. [268] were the first to observe the normal modes of the coupled system, distinct from the individual resonances of the exciton and cavity photon. In their experiment, the sample was prepared with a non-flat planar surface, enabling them to change the cavity resonance frequency via probing different locations of the surface. As the cavity resonance approaches the QW resonance, the coupling constant increases, leading to a marked splitting in the reflection spectra. This mode splitting, analogous to the vacuum Rabi splitting in atomic physics, is the signature of the strong-coupling regime [266]. The magnitude of the splitting in terms of energy is proportional to the coupling strength. The results are shown in Figure 10.5. The two new normal modes, namely, LP and UP, were observed and were shown to agree well with theoretical predictions. In the same year, Tassone et al. [269] theoretically demonstrated that reflectivity measurements of the sort Weisbuch conducted offered a reliable means of detecting the lower and upper polariton modes. In the weak-coupling regime, spontaneous emission is irreversible; the excited material decays exponentially in time, losing its energy to the cavity photon, which then leaks out of the cavity, according to its own exponential decay. Photon lasers operate in the weak-coupling regime, and it is this regime in which we studied the Purcell effect (as well as β-factor) in previous chapters. In the strong-coupling regime, however, spontaneous emission is reversible. Because the coupling occurs on a faster time scale than the decay of either exciton or photon, energy is transferred between the exciton and cavity photon and the rate of spontaneous emission exhibits oscillatory behavior in the time domain [266]. Even under the strong-coupling condition, the condensation of exciton-polaritons into a highly degenerate coherent matter wave is a prerequisite for the polariton laser. Such condensation was first reported by Deng et al. in 2002 [52], and Kasprzak followed up in the same direction in 2006 [265]. It is important to mention that the type of condensation

10.2 Strong Coupling and Condensation between Quantum-well Excitons and Cavity Photons

7 quantum wells T = 5K

20 Peak position detuning [meV]

Refelectivity (A.U.)

Quantum well resonance

15

α =3.0 × 104 cm–1 N (QW) = 5

10 5 0 5 –10 –15 –15 –10

0

Figure 10.5

1.52

221

–5 0 5 10 15 Cavity detuning [meV]

20

1.64 1.56 1.60 Photon energy (eV)

Experimental results from Weisbuch et al., who first demonstrated strong coupling of exciton and cavity photon in a semiconductor microcavity. The splitting of the reflection dip (a) and dispersion of two new modes (b) are signatures of the strong-coupling regime. Reprinted from reference [268] with permission from American Physical Society (APS).

depends on the density of excitons. In the low-density limit, when the exciton-exciton separation is much greater than exciton Bohr radius, the excitons interact negligibly with each other and follow Bose statistics. Thus, in the low-density limit the exciton gas forms a BEC [270]. In the high-density limit, the particle separation is less than the Bohr radius, which leads to non-negligible interaction between the constituent electrons and holes of neighboring excitons. As a result, the gas may no longer be treated with Bose statistics. Thus, in the high-density limit, the excitons form a Bardeen-Cooper-Schriefer (BCS) condensate, similar to that which describes superconductivity [271]. These limits may be summarized in the following way: nD aB ≪1 nD aB ≫1

exciton gas; BEC; electron  hole plasma; BCS

ð10:11Þ

where nD refers to the D-dimension exciton density and aB is the Bohr radius. Table 10.1 lists several important parameters in low-density BEC systems [264]. Imamoglu et al. [50] first predicted that lasing from exciton-polariton condensates is feasible. Specifically, the authors stated that an exciton laser may be viewed as a photon laser without inversion. This statement presupposed the low-density limit of the exciton gas, where the excitons form a BEC, rather than the high-density limit, where the excitons behave like Cooper-pairs [50]. The main claim was that inversionless polariton lasing is possible when the thermal de Broglie wavelength λTph exceeds the exciton Bohr radius aB, where

222

Beyond Nanolasers: Inversionless Exciton-polariton Microlaser

Table 10.1 Parameter comparison of BEC systems. System

Atomic gases

Effective mass m*/me Bohr radius aB Particle spacing n−1/d Critical temperature Tc Thermalization time/Lifetime

Excitons

Polaritons

–1

3

10 10–1 Ǻ 103 Ǻ 1nK – 1μ K 1 ms / 1 s ≈ 10–3

10 102 Ǻ 102 Ǻ 1 mK – 1 K 1 ps / 1 ns ≈ 10–2

10–5 102 Ǻ 1μm 1 – 300 K (1–10 ps) / (1– 10 ps) = 0.1 – 10

Thermal de Broglie wavelength v. Reservoir temperature

10–5

meff = m0 meff = 0.5m0 meff = 0.1m0 λ T = h/sqrt(2πmeffkBT)

10–7 λ T (m)

aB ~ 10–8 m

10–8

10–9

0

50

100

150

200

250

300

Temperature (K) Figure 10.6

Thermal de Broglie wavelength as a function of reservoir temperature with the effective mass as a parameter.

λTph

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πℏ2 ¼ m kTph

ð10:12Þ

with m* and Tph being the exciton effective mass and the temperature of a phonon reservoir, respectively. In Figure 10.6, the thermal de Broglie wavelength is plotted for several representative effective masses. With aB on the order of 10 nm (Table 10.1), Imamoglu et al.’s condition for polariton lasing dictates that experiments are done at relatively low temperatures and/or the exciton effective mass m* is small. In fact, near the ground state, m* becomes very small for the LP mode [12, 51, 265].

10.3 Coherent Emission of Radiation by the Stimulated Scattering of Exciton-polaritons

10.3

223

Coherent Emission of Radiation by the Stimulated Scattering of Exciton-polaritons The polariton laser may be best understood in contrast to both a BEC in thermal equilibrium and the photon laser (Figure 10.3 in Section 10.1). A polariton is defined as a superposition of exciton and photon, ^ k þ vi;k ^a k ^ p i;k ¼ ui;k C

ð10:13Þ

where i represents either the lower or upper branch, k is the transverse component of the wavevector, u is weight for the exciton operator, Ĉ, and v is weight for the photon operator â. From this standpoint, the polariton laser has u > v, and the number of carriers required for threshold increases as v increases. Unlike a pure BEC, however, polariton lasing is an out-of-equilibrium phenomenon. The process by which stimulated scattering occurs and coherent, monochromatic light is emitted is perhaps most easily understood through the model of Imamoglu and Ram [272], which we briefly describe in this section. Suppose a reservoir of LP exists in thermal equilibrium with a reservoir of phonons, which has a constant temperature Tph. The reservoir of LPs consists of the distribution across all k||-states with the exception of the ground state (k|| = 0). An incoherent pump is introduced, injecting excitons into the system and causing the LP reservoir to be out of equilibrium with the phonon reservoir. For each k-state wherein the occupancy probability of LPs exceeds that of phonons, a LP will emit a phonon (be scattered by a phonon) losing energy and momentum on its way to the ground state. This process proceeds through a quantum mechanical phenomenon known as final-state stimulation [272]. The ground state of the LP thus builds up. This process is schematically illustrated in Figure 10.4(a) in Section 10.1. For a coherent state to form, the “gain” must exceed the loss, which is typically attributed to radiative LP decay of incoherent photons, X   Γphn n exc ðk Þ > n phn ðk Þ > Γloss ð10:14Þ k

The minimum number of carriers, Nexc_min, needed to establish a condensate is then governed by the inequality in Equation (10.14) and may be expressed as the integral over the product of the occupation probability and the density of states, ð∞

ð∞

Nexc min ¼ dωρðωÞn exc ðωÞ ≥ dωρðωÞn ph ðωÞ 0

ð∞



0

ℏω ¼ dωρðωÞ exp kB Tph



1 1

0

≈

2:62 λ3Tph

ð10:15Þ

224

Beyond Nanolasers: Inversionless Exciton-polariton Microlaser

where the thermal de Broglie wavelength λTph was defined in Equation (10.12). If λTph is large, then the resulting carrier density of Equation (10.15) is less than that needed to establish a population inversion, as in a photon laser, by several orders of magnitude. For example, if λTph = 50 nm, then Nexc_min ≈ 2×1016 cm−3. Using the semiclassical gain mode, the carrier density needed for inversion can be computed: in an 8-nm-thick GaAs QW, at Tph = 300 K (30 K), to be on the order of 1017 cm−3 (1018 cm−3) [272]. While this model entails the stimulated scattering of polariton by phonons, we note that polariton relaxation can be made possible not only by polariton-phonon scattering but also by polariton-electron and polariton-polariton scattering. Electron-polariton and polariton-polariton scattering have also been studied as means to establish polariton condensates [273].

10.4

Electrically Pumped Polariton Microlasers Armed with background theory and prior experimental results in the field, we now summarize the demonstrations of polariton lasing. Polariton lasers have been experimentally demonstrated under optical pumping in organic gain material and semiconductor systems, up to room temperature. In the context of this book, due to polariton laser’s association with low lasing threshold and thus low pump-consumption applications, we only list the literature results in electrically pumped polariton lasers. In 2013, Schneider et al. [51] demonstrated polariton laser operation in electrically pumped semiconductor microcavities. Even though the temperature under which the experiments were conducted was not specified, it can be assumed to be cryogenic temperature. Figure 10.7 shows the structure of the cylindrical microcavity employed with electrically contacted circular pillars of diameter 20 μm. The laser contains a highly reflective DBR made of 23 periods of GaAs/AlAs (64 nm/71 nm) and a bottom DBR of 27 periods of GaAs/AlAs as the mirrors. Between the DBRs are 4 InGaAs/GaAs QW/ barriers (8 nm/6 nm) and an intrinsic GaAs section that is one effective wavelength (281 nm) long. To measure, a magnetic field of 5 Tesla is applied in the direction parallel to the growth direction with a detector that is rotated continuously, thus creating a momentum vector simultaneously. Real-space and reciprocal-space spectroscopy and imaging are enabled via a magneto-optical setup with spectral and angular resolution of 0.05 meV and 0.05 μm−1, respectively. Emission from the sample in the form of electroluminescence data is collected with a 0.4 NA objective and directed onto the imaging spectrometer such that the proper projection plane could be directed to the entrance slit of the monochromator. Figure 10.8 presents the collected electroluminescence as a function of both energy and in-plane wavevector. The emission is featured in three excitation regimes: below threshold, above threshold, and significantly above threshold, represented by current densities of ~44 A∙cm−2, ~113 A cm−2, and, ~241 A∙cm−2, respectively. The data shown in Figure 10.8 is significant in that it demonstrates experimental evidence of the lower polariton for values below threshold.

10.4 Electrically Pumped Polariton Microlasers

225

Ring electrode p-type DBR Singlewavelength cavity layer

Active layer InGaAs QWs

n-type DBR

Figure 10.7

Schematic of quantum well microcavity polariton diode. Reprinted from reference [51] with permission from Macmillan Publishers Ltd.

UP are not represented for any of the applied injection currents, due to its thermalization behavior resulting from the high Q-factor of the device. Figure 10.8(a) and (d), however, have a distinctive curve for the theoretical dispersion of the uncoupled cavity mode (dashed curve denoted by “C”) and the corresponding lower polariton dispersion (dashed curve denoted by “LP”). As the threshold is increased, polariton lasing occurs and the light becomes increasingly more confined at k|| = 0. For values below threshold pump current, the energy is distributed nearly evenly for many k values. The comparison between the applied magnetic field of 5 Tesla and 0 Tesla does not show variation when plotted in this representation. Therefore, they are modeled as a function of energy, magnetic field, and polarization, in the preceding figures. A major research goal thus far is to have device operation ideally at room temperatures. Temperature plays an important role in analyzing the results, in that the experimental setup includes a Faraday configuration to which a magnetic field is applied. Specifically, a magnetic field of 5 Tesla is applied to probe the light-matter nature of the polaritons, thus proving polariton laser emission by Zeeman splitting, which is temperature dependent in the strong-coupling regime. Figure 10.9 records the laser light-light curve as well as Zeeman splitting, both as a function of injection current density. Three regimes of operation are evident in the lightlight curve: (i) incoherent polariton emission, (ii) polariton laser operation, and (iii) cavitymediated photon laser operation. The two transitions between these three operational regimes clearly show nonlinear behavior in the experimental input-output plots. By using a magnetic field of 5 T, the nonlinear transitions are more pronounced than the case of no field (0 T). Surprisingly, the injected current densities at the two thresholds only differ by

226

Beyond Nanolasers: Inversionless Exciton-polariton Microlaser

Figure 10.8

Angular resolved electro-luminescence and energy-momentum dispersions (k||, in-plane wavevector) in various excitation regimes. Injection current density, j, is ~44, ~113, and ~241 A∙cm−2 at 0 T. (a) Below threshold, the polaritonic system is characterized by a thermal distribution of particles. (b) High ground-state population is observed at higher injection currents. (c) At even higher pump rates, photonic lasing occurs. (d–f) Corresponding dispersions at 5 T exhibiting similar, yet more pronounced, characteristic transition features. Reprinted from reference [51] with permission from Macmillan Publishers Ltd.

Figure 10.9

Zeeman splitting and emission power as functions of injected current density. The two thresholds correspond to the polariton and photon lasers, respectively. Reprinted from reference [51] with permission from Macmillan Publishers Ltd.

10.4 Electrically Pumped Polariton Microlasers

(a)

p-contact

Polariton Emission

227

(b)

21 periods of + p doped DBR

(SI) δdoped layers 17 –3 nδ = 1×10 cm

4 periods of undoped DBR

5λ/2 cavity 4 pairs of In0.1Ga0.9As QWs

n-contact 4 periods of undoped DBR 28 periods of n+ doped DBR n-GaAd substrate

Figure 10.10

Schematic and SEM image of microcavity for polariton laser of Bhattacharya et al. Reprinted from reference [12] with permission from American Physical Society (APS).

less than a factor of 3 (77 A∙cm−2 compared to 200 A∙cm−2). Zeeman splitting of the fundamental mode is most clearly observed in regime (i) and vanishes at the second threshold, indicating that both regimes (i) and (ii) fall into the strong-coupling regime. Almost concurrently, Bhattacharya et al. [12] demonstrated a polariton laser operating at T = 30 K and with an apply magnetic field of 7 Tesla. In this work, modulation doping of layers adjacent to the multiple-quantum-well region of the cavity is incorporated. Modulation doping has the effect of introducing free carriers in a region spatially separated from the location of the exciton confinement. It is believed that these free electrons behave as scatterers to the LP, and as such, electron-polariton scattering is more efficient than phonon-polariton scattering [24]. Moreover, relaxation from large to small transverse k states can be accomplished through electron gas–polariton interaction. Figure 10.10 shows the structure used, as well as an SEM of the fabricated device. Unlike nanolasers, the aspect ratio of this device is huge, with a 200 μm diameter and a length on the order of 10 μm. The active region of the structure consists of a 2.5λ cavity between 4 pairs of 10 nm undoped In0.1Ga0.9As/GaAs QWs. Between each pair of QWs lies the modulation-doped region, with approximately 1017 cm−3 electrons from silicon dopant atoms. Similar to the device of Schneider et al., DBRs of more than 20 periods are used as high reflectivity mirrors, with gradually increasing doping profiles as one moves from the cavity to the contacts. The QWs are designed such that their location coincides with the field antinodes of the cavity mode. A magnetic field is applied in the direction parallel to the growth of the epitaxial layers. This field increases the oscillator strength of excitons by reduction of the Bohr radius and thereby increases the splitting of the cavity photon and QW exciton. Also, and perhaps more importantly, the external magnetic field enables detection of Zeeman splitting of the emitted photons, which, when observed, is a signature that the device is operating in the strong-coupling regime. Spatial coherence of the exciton-polariton condensate can also shed light on optical properties of polariton lasing. Using a Michelson interferometer setup, interference patterns as a function of the difference in the two path lengths can be captured. The measurement setup is shown schematically in Figure 10.11(a): it consists of

228

Beyond Nanolasers: Inversionless Exciton-polariton Microlaser

Magneto-cryostat

0.3 J = 25 A/cm2

sample

J = 12 A/cm2 0.5

x

0.2

20 μm

|g1|

L1 L2

–0.5

0.1 M1 beam splitter 0.0 CCD

M2 Figure 10.11

0

10

Spectrometer

20 30 Δx (μm)

40

50

(a) Schematic of setup used to measure first-order coherence function of emitted light from polariton laser sample. (b) First-order coherence as a function of path-length difference between the arms of the interferometer and (inset) recorded fringe pattern at zero path-length difference. Reprinted from reference [12] with permission from American Physical Society (APS).

a Michelson interferometer with one of the arms made a variable length via a mirror attached to a piezo-translation stage. Images are taken on a CCD when emitted light passes through each arm separately (I1, I2), as well as simultaneously (I12). The firstorder spatial coherence function is then calculated based on the imaged intensities and the difference between the lengths of the two arms of the interferometer, jg1 jð ΔxÞ∝Iinterf ¼

I12  I1  I2 pffiffiffiffiffiffiffiffi 2 I2 I2

ð10:16Þ

The first-order coherence is plotted in Figure 10.11(b), as a function of the path-length difference, Δx, between the two arms. The best first-order coherence function evaluation, when both path lengths were identical, was about 25%. Also shown in Figure 10.11(b) is a recorded interference pattern at zero path-length difference, where the amplitude of the fringe corresponds to the coherence length. In this case, the maximum fringe amplitude is about 15 μm. This may be interpreted as meaning that polaritons are correlated with one another − behave in unison with the same phase − so long as they are spatially separated by no less than 15 μm. Bhattacharya et al. [274] demonstrated room-temperature operation in electrically pumped polariton lasers in 2014. Furthermore, no magnetic field was used in this work. The device’s schematic is shown in Figure 10.12(a) and an SEM is shown in Figure 10.12(b). Figure 10.12(c) shows angle-resolved electro-luminescence spectra measured under low forward bias current, which is used to extract the polariton dispersion characteristics shown in Figure 10.12(d). The normal is along the length of the cavity and perpendicular to the DBR mirrors. Two distinctive features are observed in the spectra. A dominant peak is observed below the exciton resonance (marked as Ex at 3.42 eV), which gradually approaches the exciton energy at 3.42 eVat large angles ∼30°. A much weaker peak is identified at energies above the exciton resonance at higher

10.4 Electrically Pumped Polariton Microlasers

Figure 10.12

229

(a) Schematic representation of the GaN microcavity diode. The inset shows a scanning electron microscopy image of the SiO2/TiO2 DBR mirror on one side, (b) scanning electron microscopy image of the fabricated device, (c) angle resolved electro-luminescence spectra measured at room temperature, and (d) calculated polariton dispersion from a coupled harmonic oscillator model alongside the measured data of (c). Reprinted from reference [274] with permission from American Physical Society (APS).

angles. The two peaks are identified as the LP and UP transitions, respectively. The cavity Q is experimentally estimated to be 1911, which may be the cause for being able to observe the UP, which were not observable in previous work with higher-Q cavities (Table 10.2) [12, 51]. It should be noted that the UP transitions have energies that are resonant with continuum states within the band. Therefore, there is some degree of uncertainty regarding their origin. The measured dispersions have been analyzed by the coupled oscillator model considering the strong coupling of the cavity photon to the exciton and are shown by the solid curves in Figure 10.12(d) alongside the measured data. The cavity-to-exciton detuning δ and the interaction potential, or Rabi splitting, Ω are determined to be −7 meV, and ∼32 meV, respectively.

230

Beyond Nanolasers: Inversionless Exciton-polariton Microlaser

Table 10.2 Comparison of threshold current densities between polariton and metal-clad nanolasers.

10.5

Authors

Reference

B (T)

T (K)

Q

Jth (A cm−2)

Bhattacharya et al. Bhattacharya et al. Schneider et al. Schneider et al. Hill et al. Hill et al. Lee et al. Ding et al.

[12] [274] [51] [51] [34] [34] [36] [37]

7 0 5 0 0 0 0 0

30 300 ? ? 10 77 77 300

6050 1991 6320 6320 200 140 1500 235

12 169 77 82 28300 94000 2800 107800

Discussions The implementation of magneto-optical setup of magnetic fields of up to a few Tesla is impractical from a commercial standpoint. However, for the sake of implementing the Zeeman splitting measurement, it is crucial. Schneider et al. [51] showed that the threshold current density is not a strong function of the external magnetic field, indicating that the magnetic field is only necessary for characterization. This is further confirmed by the work in Bhattacharya et al. [274]. Finally, to understand the reported threshold current densities, it is valuable to compare them to other electrically pumped semiconductor lasers in the literature. Table 10.2 shows the comparison, including reported cavity quality factors, operating temperature, and external magnetic field, in addition to the current density. While the reported threshold densities of polariton lasers are far superior to those of the others of Table 10.2, we note that all of these lasers employ either bulk or QW gain media. Lasers using quantum dots have been reported more than a decade prior, with threshold current densities of the same order of magnitude of those of the polariton lasers, despite operating in the weak-coupling regime [11]. We conclude by stating that demonstration of polariton lasers via electrical injection in a practical environment − zero magnetic field and room temperature − opens up the possibility of realizing useful ultralow energy coherent light sources. From a scientific perspective, polariton lasers are intriguing devices and should help us better understand macroscopic quantum phenomena, such as BEC, superconductivity, and superfluidity [52, 53].

11 Application of Nanolasers: Photonic Integrated Circuits and Other Applications In the past few decades, integration has led the way in electronics, from vacuum tube to the first solid state transistor then to electronic integrated circuit (IC). It is fair to say that electronic ICs have revolutionized the field of electronics by the functional improvements and cost savings they enabled. In the world of microprocessors, we have seen tremendous increases in computational power with simultaneous decreases in cost and power consumption, resulting from integration and standardized semiconductor manufacturing processes. In terms of data speed, the global data traffic exceeded the zetabyte mark by the first decade of the twentieth-first century, due to the exponentially growing number of internet applications for stationary and mobile devices as well as the number of their users. Several new trends are driving this rapid expansion, including high-definition mobile-to-mobile video streaming, Internet of Things phenomena, and rapid cloud computing. To support such high information density, powerful processing capabilities for fast data access are required. This is important not only in data centers but also in computational research labs and supercomputing facilities, where large amounts of data have to be processed quickly. Consequently, efficient information transfer between microprocessor cores, memory, and peripherals is essential. As network traffic increases, the demand for routing optical signals (on-chip) increases by the day, and there is strong motivation for developing photonic integrated circuits (photonic ICs) to address this challenge.

11.1

State of the Art for Chip-scale Integration In this section, we look at the state of the art for chip-scale integration of photonic components – of both active and passive elements − not restricting ourselves to nanolasers at this point. Significant advances in communication technologies are vital to keep up with this progression. State of the art 40 Gb/s and 100 Gb/s data transmission protocols are supported with both copper wire and fiber optics. Theoretically, however, optical interconnects make a stronger contender for high-speed networks due to their exceptional potential for higher bandwidth, lower latency, lower energy consumption, and scalability. New fiber technologies (multi-core and hollow-core fiber) have enabled up to 100 Tb/s aggregate transmission rate in a single fiber [275]. Further, several types of multiplexing technologies offer extra leeway in bandwidth capacity enhancement

232

Application of Nanolasers: Photonic Integrated Circuits and Other Applications

[276]. The achievement of Tb/s range bandwidth capacity is of particular significance for massively parallel computing and many fundamental physics experiments and stateof-the-art reduced-latency internet applications, such as video surveillance, telemedicine, smart car navigation, and tracking [277]. Although these record numbers are shown in optical fiber, many of the bandwidth enhancement techniques, such as wavelength and mode division multiplexing, are also applicable to photonic IC, where the limits are currently being explored. Some examples include the first demonstration of the wavelength division multiplexing (WDM) compatible silicon photonic platform, with aggregate data rates up to 60 Gb/s [278] and 1×8 Mach-Zehnder WDM (de)multiplexers fabricated using a standard 90 nm complementary metal-oxide semiconductor (CMOS) process [279]. While optical interconnects already comprise the backbone of modern information processing systems, low-cost electronics will continue to support many important functions such as data transport monitoring, switching, modulation, forward error correction, and fault control. However, efficient data conversion between the optical and electrical domains relies on the seamless integration of multiple components, such as lasers, modulators, detectors, amplifiers, multiplexers, demultiplexers, and logic. Systems assembled with many discrete components need to be further aligned and interconnected with optical fibers and copper wires. For example, a single 40 channel WDM terminal node can contain more than 100 devices and components and nearly 300 fiber coupling connections. Hence, various types of losses can accrue quickly, dramatically increasing power consumption and leading to high maintenance costs for these systems. The sheer number of wires and fibers, growing in response to the everincreasing volumes of data, will soon become architecturally and ergonomically unviable. Evidently, to overcome these deficiencies, new integration platforms for efficient, scalable, and multifunctional integrated systems need to be developed. By integrating optical components and functions into large-scale photonic ICs and using semiconductor manufacturing processes, photonic ICs can address some of these challenges. In the next generation of devices and systems, photonic ICs are envisioned to be combined with electronic ICs for fast computation and data transmission speed, low power consumption, and small footprint. Figure 11.1 shows the roadmap of electronic IC and the early developments of photonic IC. While on-chip photonic ICs do not currently enable as much bandwidth capacity as optical fibers, they have many advantages. First, they can be mass produced at a very low cost in a standard CMOS foundry, rather than assembled from multiple parts made with various expensive compound semiconductor technologies. Second, consolidation of as many components as possible on a single chip will improve overall efficiency of information processing networks and clean up the complexity in data centers. In particular, silicon makes a great material platform for on-chip photonic ICs. Low optical loss, mature technology, great potential for miniaturization, and an extensive library of silicon-based photonic and electronic devices make silicon photonic circuits a future technology for chip-scale optical information processing. Notable advances have been made in the industry within the past few years. In 2010, IBM announced a new CMOS integrated nanophotonics technology for dense integration of electrical and

11.1 State of the Art for Chip-scale Integration

Figure 11.1

233

Roadmap of electronic IC and the early development of photonic IC.

optical devices on a silicon chip. This technology enables monolithic integration of electrical and optical circuits on the same chip via the front end of a standard 90 nm CMOS line. Over several years, IBM Research has developed a library of front-end integrated active and passive silicon devices, scaled down to the diffraction limit [280]. In 2013, Intel announced transceiver and connector modules based on silicon photonics technology, which will be able to carry 100 Gb and 1.6 Tb of data per second, respectively. The Intel connector can optically convert bits of information and send them through 64 optical fibers at once. In 2013, Skorpios Technologies and Aurrion reported the first III-V/SOI hybrid lasers, fabricated in a commercial CMOS foundry [281]. These breakthrough technologies can significantly reduce the costs of running a data center. An emerging trend of mid-infrared photonic ICs (wavelength range of 2–10 μm) takes integrated photonics beyond telecommunications and expands its applications to portable sensing, imaging, and spectroscopy. Advancements in mid-IR photonics may greatly enhance analytical capabilities of life sciences, defense, and pharmaceutical and food industries in a compact and inexpensive format. One of the difficulties this research is facing today is the strong light absorption in SiO2 above 4 μm. The standard SOI platform is suboptimal in this wavelength range and needs to be replaced by a material transparent in the mid-IR. As such, silicon-on-sapphire and silicon-on-silicon nitride are potential candidates for CMOS compatible mid-IR photonic ICs. For further reading on mid-IR photonics, the article by Roelkens et al. provides a good summary [282]. In this book, we have focused on the study of one of the key elements required to enable photonic IC − compact, energy-efficient, and robust laser sources. We studied semiconductor nanolasers as stand-alone devices, from their fundamental physical properties, numerical simulations, device fabrication, to optical characterizations. However, because nanolasers are still in their early development stage, not much work has yet been done toward integrating semiconductor nanolasers with other components

234

Application of Nanolasers: Photonic Integrated Circuits and Other Applications

on-chip. In the next few sections, we briefly investigate such integration. Two main aspects need to be considered for applying semiconductor nanolasers in on-chip photonic ICs: material system compatibility and efficient channeling of energy from device to device on-chip.

11.2

Nanolasers’ Integration with a Silicon-based Platform From the aspect of material system compatibility, the most sought-after material system to be integrated with nanolasers (and active elements in general) is silicon-based platforms, in particular, the CMOS platform that is the standard for electronic ICs. Furthermore, because most passive elements are realized on the silicon platform, realizing silicon-compatible active components is of great importance. This is an area that has attracted a lot of research efforts in academia as well as in industry. From among the available device-implementation approaches, the silicon/III-V hybrid approach has become a popular choice for implementing large-scale integrated on-chip photonic components. This is largely because the hybrid approach allows the advantages of both the silicon and III-V approaches to be exploited in a complementary way. In the hybrid approach, the favorable optical properties of silicon in the telecommunicationwavelength region and the well-established processing technologies and infrastructure of the silicon industry are effectively combined, with the active functions of integrated III-V components for light generation, detection, and modulation. In addition, by reducing the physical dimensions and mode volumes of III-V active components to the scale of a wavelength, the performance metrics of the hybrid devices have been continuously improved without significantly degrading desirable optical properties such as a high cavity quality factor Q and an effective energy confinement in the active III-V materials. Various CMOS-compatible hybrid components (not restricted to nanolasers) have been reported in the literature including electro-refractive and electro-absorptive silicon optical modulators [283], high-performing photodetectors in silicon [284], germanium [285], and graphene [286]. Light generation and amplification have also been demonstrated in indirect bandgap materials such as silicon and germanium, but building high-performance lasers with these materials is still facing considerable complications. For example, Raman lasers require an external optical pump to trigger Raman scattering in silicon and induce Raman amplification [112]. An electrically pumped Raman laser is fundamentally attainable but is a highly complex engineering endeavor [287]. Silicon and germanium can also exhibit optical gain under tensile stress due to bandgap modification, as exemplified by recent demonstrations of electrically pumped strained germanium lasers [288]. However, the lasing threshold in these devices is still in the hundreds of kA/cm2 range, compared to 1–3 kA/cm2 in III-V semiconductor laser diodes. Most III-V compound semiconductors are direct bandgap materials with gain values often in the range of several thousand cm−1 [137]. Other advantages of III-V compounds include bandgap energies that can be tuned by varying the alloy composition, as well as high carrier mobility. Unsurprisingly, III-V semiconductors remain the material of choice when high optical gain or fast electronic response is required.

11.2 Nanolasers’ Integration with a Silicon-based Platform

235

Two of the most frequently implemented heterogeneous integration solutions of III-V and silicon components are flip-chip bonding and chip-to-chip butt coupling. Both technologies are mature and reliable and allow submicron alignment precision. A good example of an efficient flip-chip bonded laser is the device demonstrated by researchers from Fujitsu Ltd [289]. The laser exhibits high wall-plug efficiency of 7.6%. In this work, high-precision flip-chip bonding technology with exceptionally low alignment error (~0.1 μm) is a crucial factor in misalignment loss reduction. The highest reported wall-plug efficiency to date (9.5%) has been achieved in a hybrid laser by researchers from Kotura Inc. and Oracles Labs [290], which utilizes an external cavity reflective semiconductor optical amplifier (SOA), butt-coupled to a silicon waveguide Bragg mirror on SOI chip. A spot size converter is also an important part of the design and is incorporated to minimize coupling losses. This approach has proven to work well for realization of high-efficiency and high-power hybrid lasers. With this cavity design (reflective SOA and waveguide mirror) and device-to-waveguide coupling architecture, Oracle Labs has reported a butt-coupled laser with waveguide coupled power of 20 mW, tuning ranges of 8 nm and 35 nm, depending on tuning mechanism, and wall-plug efficiency of 7.8% [291]. Flip-chip bonding and butt coupling are among the best technologies to achieve the record device performance when the footprint is not a concern. However, another approach will have to be required when the component features are deeply sub-micrometer. With the subwavelength footprint requirement, monolithic integration of as-grown III-V and Si materials followed by further processing of III-V/Si is a promising approach. The main advantage of monolithic integration is that alignment can be avoided altogether. Two fundamentally different approaches to monolithic integration have emerged in the pursuit of merging III-V and silicon: epitaxial growth and wafer bonding. Strong lattice mismatch between silicon and III-V compounds presents a great technological roadblock to the former [292]. Despite this difficulty, several as-grown IIIV/Si lasers have been demonstrated, for example, the first electrically pumped edgeemitting III-V nanowire laser on silicon [293]. To date, the alternative approach − wafer bonding − has yielded the best results, both in terms of scalability and laser performance. For an in-depth discussion of the wafer bonding method and wafer-bonded silicon compatible nanolasers, refer to Chapter 2, Section 2.5. The first monolithically integrated III-V/Si microdisk laser was demonstrated by Rojo Romeo et al. [294]. The authors used molecular wafer bonding to create a III-V/SiO2/Si composite with ~1.2 μm SiO2 mediating layer. At the same time, Fang et al. [114] reported a III-V/Si evanescent laser. In this case, the optical mode propagates in the silicon waveguide with its evanescent tail interacting with the III-V slab. The authors used plasma-assisted wafer bonding, a type of direct hydrophilic bonding. This was an important milestone on the way to chip-scale integrated photonic circuits, as the direct contact between III-V and Si enabled easy, low loss mode coupling between active and passive circuit elements, as well as CMOS compatibility for part of the fabrication process. This work was shortly followed by demonstrations of other types of evanescent devices and an entire evanescent photonic link with lasers, modulators, and photodetectors [295]. Some of the hybrid devices are already on their way to commercialization.

236

Application of Nanolasers: Photonic Integrated Circuits and Other Applications

The latest works on wafer-bonded hybrid lasers include a BCB bonded distributed feedback (DFB) laser [296], a tunable hybrid laser [297], an integrated fourwavelength laser array [298], a sidewall modulated DFB [299], a microlaser [300], a nanolaser [115], and a slotted feedback laser [301]. In 2014, Santis et al. [302] proposed hybrid cavities as a way to dramatically reduce linewidth in semiconductor lasers by limiting spontaneous emission noise.

11.3

Nanolasers’ Integration with Optical Waveguides The realization of room-temperature, electrically pumped nanolasers that we saw in Chapter 7 and the various approaches to realize silicon-compatible active devices that we saw in Section 11.2 take us a step closer to the implementation of nanolasers in photonic ICs − the most sought-after application for semiconductor nanolasers thus far [303]. However, one significant problem in practical integration still exists: because of the extremely small output aperture of such a cavity (typically through the bottom aperture), the radiation from the cavity diverges very rapidly, making optical coupling between the III-V cavity and integrated Si waveguides inefficient. Therefore, an efficient coupling method is highly desired for nanolasers’ insertion into photonic ICs. We briefly looked at such possible coupling for antenna-inspired nanopatch lasers in Chapter 5. In this section, we study a more general approach for coupling surface (bottom) emitting nanolasers to waveguides in a silicon-compatible manner. We use the common metalclad nanolaser as an example.

11.3.1

Far-field Engineering of Metal-clad Nanocavities Before investigating the coupling between a nanolaser and a waveguide, let us first study the radiation patterns of stand-alone nanocavities. Figure 11.2(a) describes the most simplified version of the nanolaser under consideration, which can also be implemented as an optically pumped device. It consists of an InGaAsP gain cube measuring 350 nm on all sides, a low-index dielectric cladding layer such as SiO2 that encapsulates the gain, and a metal-cladding layer surrounding the entire structure except for the bottom to allow light extraction. Figure 11.2(b) depicts a similar cavity design as Figure 11.2(a) but is adjusted for electrical pumping, in which the gain region volume stays the same as the cavity in Figure 11.2(a) but doped InP and InGaAsP regions are added to assist carrier injection. Furthermore, the InP regions are undercut to allow a tight mode confinement in the vertical direction. The structures in Figure 11.2(a) and (b) are similar to those studied in Chapter 2 and Chapter 7, Section 7.2, respectively. To study the far-field radiation pattern of these cavities, we first focus on the simplified cavity in Figure 11.2(a). First, we fix the SiO2 cladding layers above and below the gain to be 150 nm and 350 nm, respectively. Second, we look at the case when the SiO2 cladding thickness in the xy-plane is the same 150 nm (denoted by a, b1, and b2 in Figure 11.3(a)). With this configuration, a Q factor of 2010 can be achieved, which is maximal given the gain region dimension. This cavity supports the fundamental TE

11.3 Nanolasers’ Integration with Optical Waveguides

(a)

237

(b) Undercut = 90 nm

350 nm

Ag (Metal) InGaAsP (III-V)

100 nm 350 nm SiO2 (Cladding) Bottom Post

z z

y x Figure 11.2

y Radiation to bottom

x

Schematic of (a) simplified metal-clad nanocavity, which is also suitable for optically pumped device design, (b) metal-clad nanocavity suitable for electrical pumping, with the semiconductor material stack (from top to bottom) being InGaAsP, InP, InGaAsP gain, and InP. Part (a) reprinted from reference [304] with permission from Optical Society of America (OSA); part (b) reprinted from reference [305] with permission from Optical Society of America (OSA).

(b)

(a)

(c)

Ag a

SiO2

b1

b2

a y x 90º

q = 90º 60º 30º

Figure 11.3

90º

60º ϕ

30º

60º 30º

Two-dimensional view of (top): energy-density distribution (with arrows showing electric field energy flow) and (bottom): far-field radiation pattern in the –z vertical direction for cladding layer thicknesses of (a) a = b1 = b2 = 150 nm; (b) a = 60 nm, b1 = b2 = 150 nm; (c) a = 60 nm, b1 = 150 nm and b2 = 60 nm. Reprinted from reference [304] with permission from Optical Society of America (OSA).

238

Application of Nanolasers: Photonic Integrated Circuits and Other Applications

mode with a doughnut profile at telecommunication wavelength, with a mode volume of 0:26ðλ=nÞ3 and a cavity wavelength of 1454 nm. To simulate the energy-density distribution, Gaussian-shaped, time-dependent, and randomly distributed dipole sources with time durations much shorter than the lifetime of the cavity mode are placed inside the cavity. Under these conditions, these dipole sources serve to excite the cavity mode without affecting properties of the mode. To simulation the far-field radiation pattern, emission from the cavity is projected toward the bottom of the device in the –z direction. The resulting energy-density distribution and far-field radiation pattern are shown in Figure 11.3(a). Because the cavity exhibits geometrical 90ο-rotation symmetry in the xy-plane, the far-field intensity pattern shows the same symmetry (bottom panel of Figure 11.3(a)). However, such a symmetric radiation is inefficient for coupling the emitted light into an integrated waveguide. Therefore, the rotational symmetry of the mode needs to be broken if efficient coupling is desired. One can break the symmetry by deforming the gain region geometry and/or the cladding geometry. To concentrate on the effect of such deformation on radiation patterns, we focus on deforming the cladding geometry, because deforming the gain region geometry affects cavity wavelength in addition to radiation pattern. The first step in the cladding engineering is to break the cladding symmetry in one direction, for example, the y-axis. By requiring a < b1 = b2, the 90ο-rotation symmetry of the near-field mode profile is broken, and the energy is more heavily distributed along the y-axis because of the stronger SPP interaction between the gain and the metal in the y-direction. The top panel of Figure 11.3(b) depicts the mode energy-density distribution when a = 60 nm and b1 = b2 = 150 nm. If we approximate the electric field radiation by dipole approximation in the first order, then propagation constant in x-direction is larger than that in y-direction, and the far-field radiation along x-axis is dominant. The bottom panel of Figure 11.3(b) shows such far-field radiation pattern, from which we observe that 80% of the radiated power is concentrated within 60ο (φ ¼ 30ο from the x-axis). The mode volume of this asymmetrical cavity is 0:24ðλ=nÞ3 , on par with that of the symmetrical cavity; and the Q factor is 1200, which is only slight more than 50% of its symmetrical counterpart (Q = 2010), and this decrease in Q factor can be attributed to the increased metal absorption in the y-direction. Following the route of cladding engineering, an even more directional radiation pattern can be achieved if the cladding symmetry is broken in both directions. Setting the condition a ≤ b1 < b2, the energy-density distribution and far-field radiation pattern are shown in Figure 11.3(c) for the specific setting of a = 60 nm, b1 = 150 nm, and b2 = 60 nm. The electric field is still oriented primarily in the y-direction, implying that the majority of the propagation constant is still orientated in the x-direction, with the near-field intensity pattern strongly biased toward the positive x-axis. This directionally biased near-field pattern generates unidirectional radiation from the cavity with about 70% of the power radiating along the positive x-axis. This strong directional bias results from the different surface plasmonic polariton (SPP) coupling strengths between the positive and negative x-directions in the cavity, which is a consequence of asymmetric charge accumulations. The Q factor and the mode volume are calculated to be 950 and 0:26ðλ=nÞ3 , respectively,

11.3 Nanolasers’ Integration with Optical Waveguides

239

comparable with the asymmetrical cavity analyzed in Figure 11.3(b). As a general trend, the more asymmetrical the cavity becomes, the more directional the far-field radiation pattern is, and the lower the overall cavity Q factor it processes.

11.3.2

Coupling from Nanolasers to Waveguides On-chip The far-field radiation pattern engineering discussed in Section 11.3.1 can be applied to enhance bidirectional or unidirectional coupling between a nanolaser and an integrated waveguide for on-chip applications. Ideally, the nanolaser would be an electrically pumped device (such as that schematically depicted in Figure 11.2(b)) constructed on a Si-compatible platform using techniques such as those described in Section 11.2, and the waveguide would be a SOI waveguide. To be consistent with the gain region dimensions in Figure 11.2(a), the gain region for the electrically pumped nanolaser of Figure 11.2(b) is also set to be a cubic of 350 nm. Additionally, the InP height above the gain is 100 nm and below the gain is denoted g; the InP undercut is 90 nm. Figure 11.4 conceptually illustrates the various loss channels from the cavity when the nanolaser is coupled to a two-port waveguide. Here, γcoupling represents the cavity-to-waveguide coupling rate, and γrad represents the radiation rate from the cavity counting both useful coupling into the waveguide γcoupling and wasteful emission into the substrate γsubstrate. Last, the total loss rate γtotal is the sum of γrad and metal loss γmetal. In maximizing the coupling efficiency from a nanolaser to a waveguide, one aims to maximize ηcoupling = γcoupling / γrad for the desired mode. In this study, given the small aperture of the described nanolaser, it is safe to consider a single mode SOI waveguide. Another two parameters that characterize the efficiency of the system are total external efficiency ηext = γrad / γtotal and total energy efficiency ηenergy = γcoupling × ηext, respectively, both of which can be controlled by changing the bottom InP height, g. Using a = 60 nm, b1 = b2 = 150 nm (optimal parameter for bidirectional coupling to waveguide as analyzed in Section 11.3.1), Figure 11.5(a) plots ηcoupling, ηext and ηenergy’s dependence on g.

Figure 11.4

Conceptual illustration of the various energy loss channels from the cavity, including metal and radiation loss. Reprinted from reference [305] with permission from Optical Society of America (OSA).

Application of Nanolasers: Photonic Integrated Circuits and Other Applications

(a)

(b) 80

η coupling

60

ηext. (= γrad / γtot) η coupling X ηext.

40 20 0 200

Figure 11.5

400

600 g (nm)

800

1000

500 400

Threshold gain Quality factor

1800 1500 1200

300

900

200

600

100

300

Quality Factor

Efficiency (%)

100

Threshold gain (cm–1)

240

0

0 200 300 400 500 600 700 g (nm)

(a) Coupling efficiency, total external efficiency, and total energy efficiency to an on-chip, two-port waveguide; (b) threshold gain and Q factor of the cavity, with nanolaser geometry of a = 60 nm, b1 = b2 = 150 nm. Reprinted from reference [305] with permission from Optical Society of America (OSA).

The change in g also profoundly influences the threshold gain and Q factor of the mode, whose dependence on g is plotted in Figure 11.5(b). Based on the results shown in Figure 11.5, we use g = 350 nm – which produces moderately high efficiencies and moderately low threshold gain – from now on. Figure 11.6(a) depicts the schematic of an integrated nanolaser and bidirectional waveguide system, with the xy-plane cross-sectional view across the center of the gain shown in Figure 11.6(b). From Section 11.3.1, we know that bidirectional coupling efficiency can be increased if a < b1 = b2 is satisfied. Setting both b1 and b2 to 150 nm and varying “a”, the coupling efficiency ηcoupling and Q factor as a function of “a” (until a = b1 = b2) is shown in Figure 11.6(c). Because both γcoupling and γrad are proportional to the total energy stored in the cavity, ηcoupling does not depend on the cavity’s total energy. When “a” is reduced below 60 nm, the coupling efficiency reaches a maximum of 80%, thanks to the dominant x-direction propagation constant. Similar to the case analyzed in Section 11.3.1, such an increase in ηcoupling is accompanied by the minimized Q factor of 800, because of the enhanced SPP coupling with the metal-cladding and the resulting larger metal loss γmetal. Setting a = 60 nm, Figure 11.6(d) shows the electric field intensity profile on logarithmic scale along the xz-plane across the center of the cavity, indicating the amount of energy that couples into the bidirectional SOI waveguide. The inset shows the energy-density distribution along the xy-plane at the center of the cavity. Figure 11.6(e) shows the far-field radiation pattern at the bottom of the cavity (right above the waveguide) for the field in (d), which is similar to the field pattern seen in Figure 11.3(b). For unidirectional coupling, the radiation directionality can be enhanced by setting a < b2 < b1, similar to that seen in Figure 11.3(c). Figure 11.7(a–e) shows the perspective and xy-plane cross-sectional view of the integrated nanolaser and unidirectional waveguide system, as well as the coupling efficiency γcoupling and Q factor as a function of b1 with a = 90 nm and b2 = 150 nm and the electric field intensity profile on logarithmic scale along the xz-plane with b2 = 180 nm and its corresponding far-field radiation pattern at the bottom of the cavity, respectively. At the maximum coupling efficiency of

11.3 Nanolasers’ Integration with Optical Waveguides

(a)

(b)

z

2-port waveguide Ag (Metal)

241

y x a < b1 = b2

InGaAsP (Cavity) (350 nm × 350 nm × 350 nm)

a

waveguide

InP (Post)

b2

b1

y

g

waveguide

a

SiO2 (Cladding) SOI waveguide

x

SiO2 (Bonding)

(d)

100

2000

80

1600

60

1200

40

800 400

20 Coupling

Coupling

0

(e) x z

Quality factor

Coupling efficiency (%)

(c)

x

y 90º 60º 30º

SOI waveguide 0

40

Figure 11.6

60

80 100 120 a (nm)

140

(a) Schematic of an integrated nanolaser and bidirectional waveguide system. (b) The xy-plane cross-sectional view across the center of the gain. (c) Coupling efficiency γcoupling and Q factor as a function of “a” with b1 = b2 = 150 nm. (d) Electric field intensity profile on logarithmic scale along the xz-plane with a = 60 nm. Inset: energy-density distribution along the xy-plane at the center of the cavity. (e) Far-field radiation pattern at the bottom of the cavity (right above the waveguide) for the field in (d). Reprinted from reference [304] with permission from Optical Society of America (OSA).

86% when b1 = 180 nm, the Q factor is relatively high at 1350, and we observe that both parameters are higher than those at the optimal condition for the bidirectional coupling. Furthermore, when b1 exceeds 180 nm, the total area of the cavity contacting the waveguide underneath becomes larger; thus, the radiation rate increases while the Q factor lowers. When this happens, the coupling efficiency tends to remain constant, ranging between 85% and 90%. This indicates that the energy is still efficiently flowing into the waveguides rather than leaking into the substrate. To conclude this section, we note that cladding engineering (e.g., SiO2 thickness a, b1, and b2) is only one of the methods to control the radiation pattern. Other methods are also viable, for example, the distance between the cavity and the waveguide (e.g., bottom InP post height g), the dielectric cladding thicknesses (e.g., SiO2 thickness a, b1, and b2), and the gain region geometry are the parameters that can be adjusted as well.

242

Application of Nanolasers: Photonic Integrated Circuits and Other Applications

(a)

(b)

z

1-port waveguide y x a < b2 = b1 InGaAsP (Cavity) (350 nm × 350 nm × 350 nm)

a b1

y

g

SOI waveguide

100

2000

80

1600

60

1200

40

800

waveguide

a

x

(d)

(e) y

400

20

Quality factor

Coupling efficiency (%)

(c)

b2

x

z x

90º 60º 30º

SOI waveguide

Coupling 0

0 90

120 150 180 210 240 270 b1 (nm)

Figure 11.7

(a) Schematic of an integrated nanolaser and unidirectional waveguide system. (b) The xy-plane cross-sectional view across the center of the gain. (c) Coupling efficiency γcoupling and Q factor as a function of b1 with a = 90 nm and b2 = 150 nm. (d) Electric field intensity profile on logarithmic scale along the xz-plane with b2 = 180 nm. Inset: energy-density distribution along the xy-plane at the center of the cavity. (e) Far-field radiation pattern at the bottom of the cavity (right above the waveguide) for the field in (d). Reprinted from reference [304] with permission from Optical Society of America (OSA).

11.3.3

Coupling from Waveguides to Nanocavities On-chip Thus far we have focused on the transferring of emitted light from nanocavities to waveguides on-chip, which is of crucial importance in applying nanolasers to on-chip applications. But what about the on-chip coupling of incoming light from waveguides to nanocavities? This is particularly relevant in integrating nano-modulators and nanodetectors with waveguides. Generally, if a cavity couples with a waveguide in the weakevanescent-coupling regime, the amount of energy in the cavity depends on the ratio of the radiation and absorption rates of the cavity mode to the cavity-to-waveguide coupling efficiency ηcoupling [200]. Similar to the procedure we took in studying the coupling from nanolasers to waveguides, we first look at the coupling efficiency ηcoupling and cavity-Q factor as a function of the bottom InP height, g, using the same geometries as those in Figure 11.6 and Figure 11.7.

11.3 Nanolasers’ Integration with Optical Waveguides

(b) 80 60

103

40

ηuni ηbi

20 0

200

Figure 11.8

300

400 g (nm)

500

Quni Qbi 600

102

Decay rate (THZ)

104

100

Quality factor

Coupling efficiency (%)

(a)

243

γrad, uni γrad, bi

100

γabs, uni γabs, bi

10–1

10–2 200

300

400 g (nm)

500

600

(a) Coupling efficiencies ηuni and ηbi and quality factors Quni and Qbi as a function of g for structures in Figure 11.6(a) and Figure 11.7(a), respectively. (b) Radiation rates γrad,uni and γrad,bi and absorption rates γabs,uni and γabs,bi of cavity modes in panel (a) as a function of g, respectively. Reprinted from reference [304] with permission from Optical Society of America (OSA).

As before, two conditions are considered: the first is the bidirectional waveguide case with a = 60 nm, b1 = b2 = 150 nm, where the coupling efficiency and Q-factor are denoted by ηbi and Qbi; the second condition is the unidirectional waveguide case with a = 90 nm, b1 = 180 nm, and b2 = 150 nm, where the coupling efficiency and Q-factor are denoted by ηuni and Quni. η’s and Q’s dependence on g is plotted in Figure 11.8(a) for both cases. Over the range of g values considered, ηuni and ηbi remain almost constant between 82% and 87% and between 77% and 79%, respectively. This means that γrad and γcoupling increase with comparable rates, thus resulting in almost constant γcoupling/γrad for both cases of coupling. As for the evolution of Q factor, its increase with increasing g can be attributed to the improved mode confinement and therefore decreased γsubstrate. Figure 11.8(b) plots the radiation rate γrad and the cavity-to-metal absorption rate γabs as a function of g for the two cases of coupling. We observe that while the γabs remains mostly constant because of the fixed cladding thickness along x- and y-directions, γrad strongly depends on g because of γsubstrate’s dependence on g. Based on the information revealed by Figure 11.8, coupled-mode theory can be used to analytically express the total energy (|ψbi|2 or |ψuni|2 for bidirectional and unidirectional coupling, respectively) in the cavity when light couples from the integrated waveguide [306]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dψ 1 ¼ jωi ψi  γrad;i ð gÞ þ γabs;i Þψi þ ζ γwaveguide;i ð gÞ  sinput dt 2

ð11:1Þ

Here, the subscript “i” refers to either bidirectional or unidirectional coupling. The quantity ωi is the cavity mode’s frequency, and |sinput|2 represents the input power from the waveguide. The constant ζ is 1/2 for the bidirectional case and is unity for the unidirectional case. From Figure 11.8, we know that γrad;i ðgÞ and γwaveguide;i ð gÞ vary with changing g, while γabs;i ð gÞ and the coupling efficiency ηi are constant with

244

Application of Nanolasers: Photonic Integrated Circuits and Other Applications

changing g. Under these circumstances, the steady-state total cavity energy |ψi|2 at the resonant frequency ωi is jψi j2 ¼ 4ηi ζ  Equation

(11.2)

shows

that

γrad;i ð gÞ γrad;i ð gÞ þ γabs;i the

total

2 jsinput j energy

2

ð11:2Þ is

proportional

to

γrad;i ðgÞ=ðγrad;i ðgÞ þ γabs;i Þ2 and reaches a maximum when γrad;i ð gÞ ¼ γabs;i . Reading from Figure 11.8(b), the optimal g value for the bidirectional and unidirectional cases are 430 nm and 380 nm, respectively. The maximum energy in the cavity under the optimized condition is thus jψi j2 ¼ ðηi ζ =γrad;i Þjsinput j2 , which is proportional to the coupling efficiency and inversely proportional to the total cavity-to-radiation rate. Taking g = 400 nm, a near-optimal value for both coupling scenarios, Figure 11.9(a) depicts the electric field intensity distribution on a logarithmic scale in the xz-plane when light is input from the left side of the waveguide. If the input field has 10 mW of power, Figure 11.9(b) shows the simulated total energies in the cavity for both cases of coupling, with the maximum energy being 3.0 fJ for bidirectional coupling and 7.7 fJ for unidirectional coupling, respectively. Even though the maximum energy for unidirectional coupling is more than twice that for bidirectional coupling, the main contributor to the difference is the difference in the constant ζ in the two cases, and a minor contributor is the slight difference in coupling efficiency (ηuni > ηbi). In summary, in Section 11.3 we looked at methods to engineer the far-field radiation pattern, maximizing nanolaser-waveguide coupling efficiency as well as cavity energy in integrated systems by employing methods such as cladding engineering and the control of semiconductor pedestal height.

Figure 11.9

(a) Profile of |E|2 on logarithmic scale along xz-plane for unidirectional and bidirectional cases for g = 400 nm when light is coupled from waveguide. (b) Total energies |ψuni|2 and |ψbi|2 inside cavity for 10 mW waveguide input as a function of g. Reprinted from reference [304] with permission from Optical Society of America (OSA).

11.4 High-speed Optical Communication with Nanoscale Light Sources

11.4

245

High-speed Optical Communication with Nanoscale Light Sources In various aspects of photonics applications, in particular for short-reach applications such as data swapping, processing among computers, chip-to-chip and intra-chip interconnects, the demand for high bandwidth processing is ever increasing. As current electronics components face their bandwidth bottleneck, the use of photonics for high-bandwidth applications has been proposed and investigated over the past few decades. In a typical semiconductor laser, as the pumping level increases, the rise in modulation resonance frequency is accompanied by an increase in damping in communication links. As a result, the maximum bandwidth is limited. To combat this, techniques such as optical injection locking has been applied to enhance the resonance frequency. In an optical injection locked system, the system’s damping may actually decrease with increasing resonance frequency, thus allowing efficient modulation at higher frequencies than that of the stand-alone laser. This technique has been applied to micro-scale vertical-cavity surfaceemitting lasers (VCSELs), and modulation bandwidth exceeding 100 GHz has been shown experimentally [307]. However, such optical injection locking schemes typically require an external laser source to lock the microlaser; therefore, the system cannot be chip-scale integrated. Nano-emitters (both nanoLEDs and nanolasers) have since been proposed for high-speed modulation applications by using their intrinsic bandwidth [8, 9]. In addition, nano-emitters have lower power consumption compared to their micro-sized counterparts. Nano-emitters owe such intriguing features to the enhancement of spontaneous emission rate into the cavity mode, also known as the Purcell effect when compared with the rate of spontaneous emission into free space (see Chapter 3). In this chapter, we study the modulation response of nanolasers.

11.4.1

Small-signal Modulation Dynamics To obtain the modulation response of nanolasers and nanoLEDs, rewriting the rate equations (Equations (1.7) and (1.8) in Section 1.1 of Chapter 1) and adding a small signal term to carrier density n, injection current I, and photon density S, we have ∂n I ¼ ηi  Rnr ðnÞ  Rsp ðnÞ  Rst ðnÞS; ∂t qVa ∂S S ¼  þ ΓE βðnÞRsp ðnÞ þ ΓE Rst ðnÞS ∂t τp with

ð11:3Þ

n0 →n0 þ ΔnðωÞ I0 →I0 þ ΔiðωÞ S0 →S0 þ ΔsðωÞ

Here the subscript “0” denotes the steady-state solution, and the “Δ” in front of the parameters denotes small signal. Following the small signal analysis detailed in many textbooks (e.g., [137], pp. 195–201), one can obtain the modulation response function: M ðωÞ

246

Application of Nanolasers: Photonic Integrated Circuits and Other Applications

M ðωÞ ¼

ηi ΓE vg g0 S0 =ðqVa Þ ω2r  ω2  jγω

ð11:4Þ

and the normalized modulation response function H ðωÞ: H ðωÞ ¼

M ðωÞ ω2r ¼ 2 M ð0Þ ωr  ω2  jγr ω

ð11:5Þ

In Equations (11.4) and (11.5), g is the gain coefficient, ωr is the relaxation angular frequency, and γr is the damping factor. They are expressed as g ð nÞ 1 þ αS



 vg gðn0 ÞεS0 βRsp 1 1 1 2 ω r ¼ ΓE þ þ þ τnr;Δn τsp;Δn τ0 sp;Δn S0 ð1 þ εS0 Þ2

gðn; S Þ ¼

þ γr ¼

g0 ðn0 Þvg S0 1 þ 0 ; τp ð1 þ εS0 Þ τp τ sp;Δn 1

τnr;Δn

þ

1 τsp;Δn

þ

g0 ðn0 Þvg S0 vg gðn0 ÞεS0 βRsp þ ΓE þ ΓE 2 1 þ εS0 S0 ð1 þ εS0 Þ

ð11:6Þ

where α in the gain coefficient expression is the nonlinear gain suppression coefficient. The various lifetimes and their derivatives are defined as 1 ¼ A þ 3Cn20 τnr;Δn ∂Rsp ðnÞ 1 at n ¼ n0 ¼ τsp;Δn ∂n

ð11:7Þ

∂Rsp;k ðnÞ 1 at n ¼ n0 ¼ τ0 sp;Δn ∂n From Equation (11.7), τ0 sp;Δn is seen to decrease as Rsp;k ðnÞ increases, as a result of the more pronounced Purcell effect in nanolasers compared to larger lasers. This is the source of the change in modulation bandwidth of nanoscale light sources. Figure 11.10 shows an electrically pumped, Si-compatible metal-clad nanocavity design. Depending on the device radius, fabrication accuracy, and operating temperature, this device serves as either a nanoLED or a nanolaser. After InP substrate removal and bonding to Au-coated Si substrate, the device can be fabricated using standard topdown fabrication techniques. The n-InP, active medium, and p-InP thicknesses are 30, 160, and 30 nm, respectively. It is encapsulated in SiNx (for electrical insulation and passivation) and Ag (for mode confinement). Compared to the devices studied in Chapter 2 and Chapter 7, the present structure has thinner semiconductor layers, thus facilitating even smaller mode volume at the expense of lowered Q factor. As we learned

11.4 High-speed Optical Communication with Nanoscale Light Sources

247

Light output Ag n-InP

n-InGaAsP

Active material SiNx

220 nm Ag

p-InP

Au Si

Figure 11.10

Schematic of an Ag metal-clad nanocavity resting on Si (after InP substrate removal) behaving either as a nanoLED or a nanolaser. The device is designed for electrical pumping, with 30 nm n-InP, 160 nm active medium, 30 nm p-InP, with a thin SiNx layer sandwiched between the semiconductor and metal-cladding for electric insulation and passivation. Reprinted from reference [9] with permission from Optical Society of America (OSA).

in Chapter 3, the lowering of Q factor plays a lesser role than the lowering of the mode volume in the Purcell effect – the effect that is positively related to the modulation bandwidth. We first investigate the modulation bandwidth of this structure with three different gain media: bulk, multiple-quantum-well (MQW), and quantum dot (QD). The bulk material is In0.53Ga0.47As, customary to electrically pumped nanolasers. The MQW is strain-compensated 20 pairs of InGaAsP/InGaAlAs, utilizing the higher differential gain provided by the more symmetrical conduction and valence bands from the strain effect as well as the high confinement inherent to MWQ structure. This gain material has been shown to have 30 GHz bandwidth in edge emitting lasers [308]. For InxGa1-xAs QD gain medium, uniform sized, submonolayerdeposited high-dot density of NQD = 2 × 1012 cm−2 QDs are assume [309], resulting in an effective QD height of 3 mm and radius of 2.5 mm. The surface recombination lifetime of the QD is taken to be 0.05 ns. Other parameters of the three gain media are provided in Table 11.1. In the LED regime, the modulation bandwidth is characterized by the 3 dB bandwidth f3 dB: f3dB ¼

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2π τ2p þ τ0sp;Δn

ð11:8Þ

Both τp and τ0 sp;Δn in the modulation bandwidth expression of Equation (11.8) are functions of the Q factor. As Q increases, τp increases, but on the other hand, τ0 sp;Δn

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Table 11.1 Parameters used in the analysis of modulation response of a metal-clad nanocavity. Parameter

Bulk

Quantum well

Quantum dot

Wavelength λ (nm) Optical confinement factor ΓE Layer number count Thickness of a single active layer (nm) Spontaneous emission factor β Surface velocity vs (104 cm∙s−1) Gain suppression coefficient α (10−17 cm3) Auger coefficient C (10−30 cm6∙s−1) Half-width-at-half-maximum (HWHM) of homogeneous broadening Γcv = R(ω-ωr)/2 Gaussian linewidth parameter from inhomogeneous broadening σ (meV)

1542 0.6 1 160 0.41 4 2.3 1 20

1550 0.2 20 6 0.41 4 2.3 1 20

1265 0.044 10 3 0.41 − 1 1 10,20

−

−

5

Reprinted from reference [9] with permission from Optical Society of America (OSA)

Figure 11.11

(a) τp and τ0 sp;Δn as a function of Q. (b) τp and

α vg g0 ðn0 Þ

as a function of Q for different gain medium.

Reprinted from reference [9] with permission from Optical Society of America (OSA).

decreases until saturation is reached. Saturation of τ 0 sp;Δn occurs when cavity linewidth is much narrower than the inhomogeneous broadening linewidth, under the same condition as the Purcell factor saturation we studied in Chapter 3. This dependence of τp and τ0 sp;Δn on Q is plotted in Figure 11.11(a). In the lasing regime, the maximum bandwidth is determined by the 3 dB maximum relaxation frequency under the condition of 2ω2r ¼ γ2r [137]: fr;max ¼

2 τp þ

pffiffiffi 2 α 0 vg g ðn0 Þ



ð11:9Þ

When Q is lower, τp is smaller; therefore, a higher material gain is necessary in order to satisfy the lasing condition. Along with the higher material gain is the smaller differential gain g0 ðnÞ, which can balance out and even outweigh the smaller τp in Equation (11.9). Figure 11.11(b) plots the denominator of Equation (11.9) τp and

11.4 High-speed Optical Communication with Nanoscale Light Sources

Figure 11.12

249

Modulation bandwidth of metal-clad nanoLEDs and nanolasers (separated by the vertical dashed line) as a function of Q factor and the normalized optical mode volume Vn with (a) bulk gain and (b) MQW gain. The scale bars show the bandwidth with unit of GHz. Reprinted from reference [9] with permission from Optical Society of America (OSA).

α ’s dependence on Q for the three gain media that have different differential vg g 0 ð n0 Þ gains. Similar to nanoLEDs, τplimits the bandwidth at high Q. In addition to Q, the modulation bandwidth is also dependent on the mode volume Veff through τ0 sp;Δn for nanoLEDs and g0 ðn0 Þ for nanolasers, respectively. Figure 11.12 shows the modulation Veff bandwidth’s dependence on both Q and the normalized mode volume Vn ≡ for ðλ0 =2nÞ3 bulk and MQW gain media, in both the nanoLED and nanolaser regime calculated based on Equations (11.8) and (11.9). Examining the nanoLED regime first, we observe that as the mode volume becomes extremely small (Vn < 0.01), f3 dB is greatly enhanced when Q factor is low and moderate, due to the larger βRsp ðnÞ. However, when βRsp ðnÞ is so large that the rapid recombination of carriers near the mode frequency ωr may deplete the carrier population at a speed faster than it is replenished by intraband scattering, a phenomenon known as “spectral hole burning” occurs. Under this circumstance, the material gain decreases significantly with increasing bias, and the device cannot reach threshold even if Q is further increased. Therefore, so long as Vn < 0.01, the device remains as a nanoLED regardless of the Q factor. However, comparing bulk and strained MQW, because QWs have more symmetrical conduction and valence bands, the gradient in the increase of βRsp ðnÞ is higher than in bulk gain, making QW nanoLEDs having a higher bandwidth but at the time more sensitive to spectral hole burning. To alleviate the effect of spectral hole burning in MQW nanoLEDs, Vn is extended to 0.02 for low Q in Figure 11.12(b).

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Application of Nanolasers: Photonic Integrated Circuits and Other Applications

Figure 11.13

NanoLED L-I curves plotted in terms of (a) total output power for Q = 250 with different Vn, (b) spontaneous and stimulated emission power for Q = 250 with different Vn, (c) total output power for Vn = 0.006 for different Q, (d) spontaneous and stimulated emission power for Vn = 0.006 for different Q. Reprinted from reference [9] with permission from Optical Society of America (OSA).

Overall, greater than 200 GHz modulation frequencies can be achieved in nanoLEDs with extremely small mode volume. Typically, MQW gain allows higher bandwidth than bulk gain, because MQW has a greater g 0 , leading to a larger fr,max in Equation (11.9). This difference in g 0 also leads to the optimal Q values being 875 and 750 for bulk and MQW gain, respectively. Figure 11.13 shows cavity Q’s and mode volume’s effect on the single mode spontaneous emission rate, and in turn, on the nanoLED L-I curve. With a fixed Q factor of 250, it is evident from Figure 11.13(a) that small Vn enhances the total output power. From the separately emitted power into spontaneous and stimulated emission shown in

11.4 High-speed Optical Communication with Nanoscale Light Sources

251

Figure 11.13(b), we see the opposite effect of Vn on the respective emission: the spontaneous emission power increases as Vn decreases because of the Purcell effect, while the stimulated emission power decreases as Vn decreases because spectral hole burning is more prominent in devices with smaller mode volume. Figure 11.13(c) and (d) shows the same analysis as a function of Q at a fixed Vn of 0.006. Comparing Figure 11.13(b) and Figure 11.3(d), we can see how Q and Vn affect the single mode spontaneous emission differently: the curves for Q of 1000 and 10000 Figure 11.13(d) completely overlap, showing the single mode spontaneous emission rate’s independence of Q when the mode linewidth defined by Q becomes much narrower than the homogeneous and/or inhomogeneous linewidth as studied in detail in Chapter 3. In conclusion, in designing a nanoLED suitable for high-speed modulation, one should aim for a cavity Q between 100 and 1000 in addition to a Vn that is as small as possible. Having identified the regime of device geometrical parameters for high-speed nanoLED and nanolaser design, we study the modulation response as well as emission characteristics. Figure 11.14(a) shows the modulation response, and Figure 11.14(b) shows the spontaneous and stimulated emission rates of a nanoLED with Q = 875 and Vn = 0.006. Similar to Figure 11.14(a–b), Figure 11.14(c–d) shows characteristics of a nanolaser with the same Q as the nanoLED (Q = 875) but with a larger Vn of 37. We see that for nanoLEDs (Figure 11.14(a–b)), spontaneous emission dominates and the bandwidth is limited by βRsp ðnÞ and τph. The bandwidth first increases with increasing injection current up to 2.4 times the transparency current (Itr), but it decreases when the current is further increased to 3 times Itr as a result of the more dominant damping at higher currents. For nanolasers (Figure 11.14(c–d)), stimulated emission dominates and the bandwidth is limited by g 0 and τp. We also observe the evolution of the modulation response as a function of injection current: from a prominent relaxation resonance at low injection levels to a flat band when 2ω2r approaches γ2, and to a damped response when 2ω2r becomes larger than γ2. For metal-clad nano-emitters with QD as the gain medium, because of the high QD density, high confinement factor provided by the metal and the assumed uniform dot size, ΓE, can be sustained sufficiently high. Therefore, they can achieve lasing even with a relatively small Q. Because of the 3D confinement of carriers in QDs, the higher differential gain g 0 makes the 3 dB maximum relaxation frequency fr,max in Equation (11.9) larger than fr,max for bulk or MQW lasers. These features are reflected in Figure 11.15. Unlike nano-emitters with bulk or MQW gain, QD nano-emitters behave like lasers and sport a uniform bandwidth over a large range of Vn from 0.001 to 100. For Γcv = 10 meV, the maximum bandwidth is as high as 320 GHz. Figure 11.15 also compares the bandwidth under two homogeneous broadening Rðω  ωr Þ ≡ Γcv conditions: because a larger Γcv reduces gain as well as differential gain, a smaller bandwidth is seen for Γcv = 20 meV (Figure 11.15(b)). Only when Q is extremely small, the gain is close to its maximum, resulting in a differential gain that is smaller than that in larger Q cavities. This is reflected in the bottom part of Figure 11.15. Note that in order to efficiently utilize the high QD gain, it is essential that the cavity resonance frequency

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Application of Nanolasers: Photonic Integrated Circuits and Other Applications

Figure 11.14

(a) Modulation response and (b) spontaneous and stimulated emission for a nanoLED with Q of 875 and Vn of 0.006. (c) Modulation response and (d) spontaneous and stimulated emission for a nanolaser with Q of 875 and Vn of 37. The device design is based on that in Figure 11.10. Reprinted from reference [9] with permission from Optical Society of America (OSA).

coincides with the QD transition frequency. In high Q conditions, similar to nanolasers with bulk or MQW gain, τph limits the modulation bandwidth, and the bandwidth of all types of nanolasers converge to 40–50 GHz (Figure 11.12 and Figure 11.15). Last but not least, under the small-signal modulation condition, an important figure-ofmerit to evaluate the energy efficiency is the power-to-bandwidth ratio (PBR), defined as the total power consumption divided by the 3 dB bandwidth ω3 dB, which refers to f3 dB in the case of nanoLED and fr,max in the case of nanolaser,   Iinj Efn  Efp Pin;CW ¼ ð11:10Þ PBR ¼ ω3dB =2π ω3dB =2π

11.4 High-speed Optical Communication with Nanoscale Light Sources

Figure 11.15

253

Modulation bandwidth of metal-clad nanolasers with QD as the gain medium with the homogeneous broadening HWHM Γcv of (a) 10 meV and (b) 20 meV. The scale bars show the bandwidth with unit of GHz. Reprinted from reference [9] with permission from Optical Society of America (OSA).

where Efn and Efp are respectively the Fermi levels of the conduction and valence bands. The analysis thus far has ignored thermal effects on modulation bandwidth. Although low Q cavity is preferred for high bandwidth operation, a high carrier density is required. Therefore, thermal consideration in a high-speed modulation setting, similar to thermal management in a nanolaser design (Chapter 8), can hugely affect device and system performance. Typically, the Fermi distribution function and spontaneous emission linewidth broaden as temperature rises, and the material gain and photon density decrease. Accordingly, the temperature rollover occurs at relatively low bias. However, for highspeed modulation, a few times the transparency current in the case of nanoLED or more than 10 times the threshold current in the case of nanolaser are needed. Therefore, one should not expect the experimentally achievable modulation bandwidth to be as high as those theoretically predicted in this section. In order to alleviate the thermal effects, one approach is to improve the energy confinement ΓE such that the required carrier density can be relatively low even in low Q cavities; another approach is to use a gain material that is relatively insensitive to temperature, for example, QD gain.

11.4.2

Large-signal Modulation Dynamics Although the study of small-signal modulation response reveals modulation characteristics of the intrinsic speed of a device, the more relevant situation is modulation under large digital signals. This is especially the case for nano-emitters, because bit-error rate (BER) – an important figure-of-merit in digital modulations – is expected to be worse in nanolasers than in their larger-scale counterparts due to their lower output power and

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Application of Nanolasers: Photonic Integrated Circuits and Other Applications

Figure 11.16

(a) Photon number variation of a nanolaser under 1 ns width pulsed current injection (dashed line) from Ioff at threshold current Ith, to Ion at 5 x Ith. (b) Noise spectral density of output power from a nanolaser with radius of 200 nm under an injection current density of 100 kA/cm2. Reprinted from reference [77] with permission from John Wiley & Sons, Inc.

larger laser noise. Even though energy efficiency as measured by the energy data-rate ratio (EDR) has become an important aspect of the laser performance and record energy efficiency has been achieved in VCSELs [310], there is a lack of general understanding of how energy-data efficiency is related to device size and modulation bandwidth and other performance characteristics. In this section, we study the modulation bandwidth, among other parameters, of nanolasers under large-signal modulation. In large-signal modulation, the maximum data rate is governed by the laser turn-on and -off delay time as well as the BER, which determines signal quality. Figure 11.16(a) shows a typical semiconductor metal-clad nanolaser’s photon number’s, Nph’s, evolution under a 1 ns square-shaped pulsed current, when the current is increases from its off-state at Ioff ≈ Ith to on-state at Ion ≈ 5  Ith . We see that Nph first experiences a delayed turn-on (characterized by τon) at the rising edge, which is followed by an overshoot and oscillations until Nph stabilizes to the on-state Nph,on. When the current is decreased to its off-state, Nph drops to Nph,off over the turn-off time duration τoff, where   τoff is defined as the time that it takes Nph to drop to 0:1  Nph;on  Nph;off above Nph,off. For non-return to zero (NRZ) format, because there is at most only one rising or falling edge in a data period, the longer time of τon and τoff is set as τd. The BER for signals of NRZ format is expressed as [58]

 Pon  Poff 1 Qs BER ¼ erfc pffiffiffi with Qs ≡ ð11:11Þ 2 σon þ σ off 2 where Pon and Poff are the on- and off-state output power, and σon and σoff are the on- and off-state noise power. BER characterizes the noise performance and is an important figure-of-merit in evaluating the laser performance when it is used for digital modulation.

11.4 High-speed Optical Communication with Nanoscale Light Sources

255

The output noise power σ can be calculated by incorporating Langevin’s noise sources into the rate equations (1.7) and (1.8) of Chapter 1, which now become ∂n I ¼ ηi  Rnr ðnÞ  Rsp ðnÞ  Rst ðnÞS þ Fn ðtÞ; ∂t qVa ∂S S ¼  þ ΓE βðnÞRsp ðnÞ þ ΓE Rst ðnÞS þ Fp ðtÞ ∂t τp

ð11:12Þ

where Fn(t) and Fp(t) are the Langevin’s noise sources for carriers and photons, respectively. Following procedures detailed in textbooks [137], the photon number noise spectral density S(ω) can be calculated. In the simplified case where the output partition noise is ignored; S(ω) and output power noise spectral density SP(ω) are related by

SP ðωÞ ¼

ℏωr τrad

2 S ðωÞ

ð11:13Þ

In Equation (11.13), photon radiation lifetime τ rad ¼ Qrad =ωr , where Qrad is the radiation Q factor, is determined by the output power coupling. Figure 11.16(b) plots SP(ω) of an example electrically pumped cylindrical nanolaser with a core radius of 200 nm (similar in design to that in Section 7.3 of Chapter 7), when it is operated above threshold at 100 kA/cm2 current density. With the knowledge of SP(ω), the output noise power σ within a 2Δω bandwidth is σ¼

1 2π

ð Δω Δω

SP ðωÞdω

ð11:14Þ

Inserting Equation (11.14) into the Qs expression in Equation (11.11), one can then obtain the noise-limited maximum modulation bandwidth of a laser by searching the maximum integration limit Δω, which reaches a minimally required Qs, and by searching the noise-limited date rate Dn = 2Δω. For example, a 10−12 BER is required for today’s typical 10 Gb/s network, which corresponds to a signal quality of Qs = 7.13. In this case, one searches for Δω such that Qs ≥ 7.13. Both the turn-on and -off delay time τd and the minimum time that the signal remains in any given state to reach the required BER τn ¼ 1=Dn affect the minimum data period. The final data rate for a laser in digital modulation is expressed as DR ¼

1 τd þ τn

ð11:15Þ

Under NRZ format, the average energy consumption per bit, also known as EDR, is   0:5 Pin;on þ Pin;off ð11:16Þ EDR ¼ DR where Pin,on and Pin,off are input power at on- and off-states, respectively.

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Application of Nanolasers: Photonic Integrated Circuits and Other Applications

11.5

Silicon-compatible Miniature Laser While existing work on a Si-compatible nanolaser and its coupling to waveguide is limited and largely theory based, much more work has been done in Si-compatible micro-scale lasers. In this section, we review recent progress in III-V/Si microlasers, in particular, distributed feedback (DFB) lasers, for compact, high-density integration with silicon photonics, as well as the problems yet to be addressed for their practical implementation in chipscale integration. These works shed light on the design and realization of Si-compatible nanolasers, as well as their integration with the existing silicon photonics framework.

11.5.1

Optically Pumped Sidewall-modulated III-V/Si DFB Microlaser In this section, we briefly discuss the optically pumped III-V/Si DFB laser design, which is realized by implementing self-aligned sidewall modulated Bragg gratings for optical feedback in order to maximize the interaction between the electric field of the mode and the available gain in the structure [299]. This approach can help reduce lasing threshold and/or physical size of the device. Using this design, Bondarenko et al. [299] showed lasing in a 100-μm-long hybrid grating with a sub-micron waveguide cross section (Figure 11.17(a)).

Figure 11.17

(a) Schematic of a sidewall modulated DFB laser, top view; simulated (b) Ex component and (c) Ez component of the fundamental TE-like mode for a 500-nm-wide and 550-nm-tall III-V/Si waveguide with 250 nm silicon layer, 300 nm InGaAsP layer; (d) complex space of resonant condition solutions for four forward and four backward propagating longitudinal modes. The solutions represent the lasing threshold for single section DFB lasers without facet reflections. Parts (a)−(c) reprinted from reference [312] with permission from Optical Society of America (OSA).

11.5 Silicon-compatible Miniature Laser

257

The waveguide sidewall modulation is achieved by varying its width with period Λ along the mode propagation direction, with w1 and w2 widths corresponding to the wide and narrow sections of the guide. In this work, the plasma-assisted wafer-bonding technique was used, which is followed by self-aligned fabrication [118, 119] for scalable, simple, and alignment-free fabrication. Note that the choice of wafer bonding method was limited by the low-temperature process requirement to be below 400 ˚C for III-V material, and the reliance of the cavity design on direct III-V/Si bond configuration. In the following, we examine only the fundamental transverse electric (TE)-like mode of the composite III-V/Si waveguide. This is justified by the use of MQW InGaAsP gain material for the device, where the TE-like mode experiences significantly higher gain than the transverse magnetic (TM)-like mode [311]. In analyzing such waveguides, the gain/mode overlap of a bulk InGaAsP/Si composite waveguide is considered for computational simplicity (Figure 11.17(b) and (c)). Numerical simulations show that the fundamental TE-like mode is evenly distributed between the silicon and III-V layers. Quantitatively, this is described by a modal confinement factors in the gain, ΓIII-V, and silicon, ΓSi, that are both close to 0.5. Estimation of the lasing threshold of DFB lasers requires knowledge of the grating coupling coefficient κ, gain volume interacting with the optical field, and losses in the system. Coupled Mode Theory (CMT) [313] is instrumental in understanding both passive and active Bragg grating behavior. For the simplest DFB laser with no reflections at the facets, using CMT, the resonance condition takes the form of [314] Δm coshð Δm LÞ þ jΔm sinh ð Δm LÞ ¼ 0

ð11:17Þ

ðΔm Þ2 ¼ ½ðΔβm Þ2 þ κ κ

ð11:18Þ

where

κ is a coupling coefficient and L is a length of DFB grating. A solution to Equation (11.17) is the propagation constant Δβm, which is a complex value. The real part, Re(Δβm), of the propagation constant corresponds to the detuning from the Bragg wavelength, while the imaginary part, Im(Δβm), corresponds to the modal threshold gain. Equation (11.17) can be solved numerically, using an iterative solver such as the Newton-Raphson method. It is useful to plot multiple complex solutions to this equation on a complex plane, as shown in Figure 11.17(d). Note that both parts of the propagation constant are normalized by κL. This plot is standard for every symmetric DFB, but κ is necessary to calculate the threshold gain. The coupling coefficient quantifies interaction between forward and backward propagating modes in the grating and generally has a transverse (κxy) component due to the non-zero Ex and Ey components of the electric field as well as a longitudinal (κz) component due to nonzero Ez component. The full coupling coefficient for the fundamental TE-like mode and its backward propagating TE-like twin is κ = κxy− κz [315]. It is appropriate to neglect coupling to higher-order modes and only consider coupling of the forward propagating mode to the counter-propagating mode of the same order and polarization. However, in this case, κz makes a weighty contribution

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Application of Nanolasers: Photonic Integrated Circuits and Other Applications

to the total κ, since the Ez component is not only large but also strongly overlaps with the sidewall grating (Figure 11.17(c)). Numerical simulation estimates the transverse component of the coupling coefficient to be 275 cm−1 and the longitudinal component to be 190 cm−1. Thus, the total coupling coefficient is 85 cm−1, taking into account fabrication imperfections. For this value of κ and a 100 μm grating length, the product κL is 0.85. We can now look for a complex propagation constant (Re(Δβm), Im(Δβm)) of the lowest order longitudinal mode (m = ±1) corresponding to this value of κL (Figure 11.17(d)). For a gain/mode overlap of ΓIII-V = 0.5 and negligible optical loss, the lowest order solution of Equation (11.17) yields a threshold gain of ~400 cm−1. The material gain in MQW media can be as high as 6000 cm−1, which is sufficient to compensate for the lower gain/mode overlap in such MQW InGaAsP/Si composite waveguides. Optical measurements on the DFB structures are carried out on a standard micro-photoluminescence setup with a 1064 nm nanosecond pulsed fiber laser. A single mode lasing peak is observed at 1515 nm wavelength with linewidth below the monochromator resolution limit of 0.35 nm. From light-light measurements (Figure 11.18), the threshold peak pump intensity can be extracted to be around 530 W/mm2. To conclude this section, we note that the flexibility of a sidewall-modulated grating design can also enable the development of lowthreshold hybrid DFB lasers, albeit at the expense of a larger physical size. Work on an electrically pumped version of the laser involves additional challenges, as we discuss in the following section.

Figure 11.18

Light-light curve of a fabricated optically pumped device (the same data in logarithmic scale – bottom right inset; SEM image of the hybrid grating is on the upper left inset). Reprinted from reference [299] with permission from American Institute of Physics (AIP) Publishing LLC.

11.5 Silicon-compatible Miniature Laser

11.5.2

259

Electrically Pumped Sidewall-modulated III-V/Si DFB Microlaser The first electrically pumped hybrid laser was based on the InAsP microdisk [294] and AlGaInAs-silicon evanescent designs [114] in 2006. Several other approaches for a better III-V/Si device-to-waveguide coupling were developed shortly after these first demonstrations [316]. Almost a decade later, the hybrid platform has become essential to the development of photonic integrated circuits providing elements on a chip such as optical amplifiers, 2 R regenerators, modulators, and photodetectors [317]. In 2014, an efficient electrically pumped DBR hybrid laser was also demonstrated by Duan et al. [318]. As described in Section 11.5.1, in the case of the optically pumped hybrid DFB laser, the active III-V layer is directly bonded to the silicon layer, and the mode of the III-V/Si composite waveguide is almost equally confined to the silicon and III-V layers. For electrical carrier injection, a p-i-n heterostructure must be incorporated into the III-V epitaxial layers. In a p-i-n heterostructure, the active region is intrinsically doped, while the top and bottom contact layers, InGaAsP and InP, are p- and n-doped, respectively. Figure 11.19 schematically depicts the semiconductor stack after wafer bonding. The top and bottom InGaAsP layers are heavily doped (1×1019 cm−3) for low resistance ohmic contacts. The InP cladding layers have a lower doping level (1×1018 cm−3) to reduce optical losses from impurities, as a small portion of the mode propagates within these cladding layers. The n-doped bottom layer (InP-n and InGaAsP-n– in Figure 11.19) is hereafter referred as the bonding layer.

Figure 11.19

Schematics of the III-V epitaxial layers bonded to Si for an electrically pumped laser. Reprinted from reference [312] with permission from Optical Society of America (OSA).

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Application of Nanolasers: Photonic Integrated Circuits and Other Applications

Figure 11.20

(a) Schematics of the simulation domains. (b) Normalized electromagnetic field for InGaAsP-n– = 125 nm and InP-n = 200 nm and (c) InGaAsP-n = 20 nm and InP-n = 20 nm. Reprinted from reference [312] with permission from Optical Society of America (OSA).

The bonding layer thickness is one of the most important optimization parameters since it affects the gain/mode overlap. Generally, a higher gain/mode overlap may be achieved with a thicker III-V layer stack. However, the mode propagation losses in the doped layers and complexity of the mode coupling to a silicon waveguide are the challenges associated with thick layers. To illustrate this problem, we consider two extreme cases in this section: a III-V stack with a thick (325 nm) and a thin (40 nm) bonding layer. The choice of a specific design depends on desired functionalities of the device. Figure 11.20(a) shows the schematic diagram of the cross-sectional geometry, and the mode distribution of the structures with the thick and thin bonding layers is presented in Figure 11.20(b) and (c) for comparison. In both cases, the width of the III-V region is set to be 500 nm to ensure that only the fundamental mode is excited. The width of the silicon waveguide is 1200 nm and its thickness is 250 nm. To reduce radiation losses, a 130 nm layer of SiO2 covers the sidewall of the III-V region. Figure 11.20(b) and (c) shows the spatial mode distribution for stacks with the thick and thin bonding layers, respectively. The bonding layer thickness is InGaAsP/InP 125/200 nm in Figure 11.20(b) and InGaAsP/ InP 20/20 nm in Figure 11.20(c). Evidently, a thicker bonding layer yields higher gain/ mode overlap and lowers the overlap with the silicon waveguide. In this case, the confinement factor ΓIII-V for the gain region is 0.17 while that for the silicon waveguide, ΓSi, is only 0.07. The spatial distribution of the electromagnetic field for a thin bonding layer is shown on Figure 11.20(c). We observe that the mode only partially extends from the silicon waveguide into the gain region. In this case, the confinement factor ΓIII-V is only 0.101,

11.5 Silicon-compatible Miniature Laser

261

while for the silicon waveguide, ΓSi, it is 0.407. The advantage of the thin bonding layer is that the mode is strongly confined in silicon, which favors the mode coupling to a passive silicon waveguide. Also, as reported by Santis et al. [302], if the energy is generated and stored in the same lossy III-V material (the gain media and the doped layers), there is an excessive spontaneous emission noise degrading the laser coherence, so the laser may not meet the requirements for phase-coherent modulation. The disadvantage of the thin bonding layer is that the low ΓIII-V can lead to a higher threshold current density compared to the thick bonding layer and high ΓIII-V. Low threshold current helps reduce power consumption and avoid heating. Thermal effects can introduce drift of the emission peak, reduce the internal quantum efficiency and device reliability, as well as accelerate its degradation [319]. Besides optimization of electrical and optical properties of the semiconductor structure, it is necessary to couple these modes to a silicon waveguide with minimal optical loss. To address this issue, a tapered waveguide design is proposed for low–loss mode coupling. We study the tapered waveguide design in the next section.

11.5.3

Coupling III-V/Si Edge-emitting Lasers to Si Waveguide Efficient light routing between passive and active components in a photonic chip requires a coupler, where the mode is pulled into a silicon waveguide from the active section. To couple edge-emitting laser light to a lateral silicon waveguide, the schematics of three taper geometries are shown in Figure 11.21. Design I is a three etching level taper used by Heck et al. [317]. Design II is a conventional taper used by Lamponi et al. [320], and Design III is a conceptual 3D taper waveguide. Each of these tapers couples the mode from the III-V region to the silicon waveguide adiabatically to prevent excitation of other transverse modes. Here Designs I and II are presented as examples of taper couplers, since they have been previously shown to exhibit high transmission efficiency due to low modal impedance mismatch [317, 320–322]. In Designs I and II, the silicon waveguide width is shorter than that of the III-V. This is achieved by first fabricating the silicon waveguides on a SOI platform, then bonding them to the III-V stack and executing top-down fabrication on the entire III-V/Si stack. To illustrate the modal behavior in different regions of the tapered waveguides, we use the thicker bonding layer in these examples. Taper Design I consists of three etching level tapers where each material layer (p-cladding, gain media, and n-cladding) is tapered with a different taper length. In numerical simulations, the tapered tip widths are set to 400 nm, although in reality this can vary from layer to layer. The silicon waveguide thickness is fixed at 1.2 μm to demonstrate that the spatial mode distribution changes during the coupling process. Note however that the silicon waveguide can be tapered in the opposite direction prior to wafer bonding of III-Vand Si to increase the coupling efficiency [317]. Figure 11.21(a)−(c) shows the evolution of the spatial mode distribution along the tapered waveguide as the mode is pulled into the silicon waveguide. The optical mode initially is localized in the gain media where laser light is generated (Figure 11.21(a)). With tapering of the III-V, the optical mode gradually transfers to the silicon waveguide. In the last step (Figure 11.21(c)), almost no

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Application of Nanolasers: Photonic Integrated Circuits and Other Applications

MQW

InP - p InGaAsP - p++

InGaAsP - p++

Si SiO2

SiO2 (c) (b) (a)

Design I

InGaAsP - p++

Si

Si

SiO2 (f) (e) (d)

Design II

(i) and (j) (h) (g)

Design III

(a) (d)

(g)

(e)

(h)

(b)

(c) (f)

Figure 11.21

(i)

(j)

Three taper designs for coupling the electromagnetic mode from the III-V/Si composite waveguide to the silicon waveguide. Parts (a)–(c), (d)–(f), and (g)–(j) are the mode profiles in different cross sections of the mentioned tapers, indicated by the dashed lines, for Designs I, II, and III, respectively. The 3D tapers were simulated on thin bonding layer, but (j) also shows the result for a thicker bond layer. The arrow indicates the direction of tapering the waveguide to follow the corresponding mode profile evolution. Reprinted from reference [312] with permission from Optical Society of America (OSA).

modal energy remains in the gain media. The total length plays an important role in this design. H. Park et al. used a three etching level taper with the total length of 80 μm in their work on the hybrid AlGaInAs-Si preamplifier and photodetector, and the authors measured the taper losses of 0.5 dB per transition region [321]. Increasing the total taper length to 160 μm, Kurczveil et al. [322] reduced the losses to 0.3 dB per transition region and applied the tapers on an integrated hybrid silicon multi-wavelength asymmetric waveguide grating laser. In the second approach, all three layers have the same taper length and tip width, as illustrated in Figure 11.21 (Design II). All dimensions are identical to Design I, unless otherwise noted. The spatial mode distributions for the different cross-section widths are shown in Figure 11.21(d)−(f). The evolution of the mode from the III-V to silicon waveguide is similar to Design I. In this configuration the silicon waveguide is also tapered on a pre-bonding fabrication step. Design II in reference [114] is extremely sensitive to the tip width, because if the tip width is not sufficiently small, the optical mode cannot be pulled into the silicon waveguide efficiently. Fortunately, a larger silicon

11.5 Silicon-compatible Miniature Laser

Figure 11.22

263

Power profile of the 3D taper with a thin bonding layer. The thicknesses of the gain media, bonding layer, and silicon waveguide are 400 nm, 40 nm, and 250 nm, respectively, and the 3D taper is 15 μm long. Reprinted from reference [312] with permission from Optical Society of America (OSA).

waveguide thickness allows a more efficient coupling with larger width III-V tapers [323]. With this design, Lamponi et al. reported 90% coupling efficiency in a 100-μm-long taper with a 400 nm taper width in the III-V region [320]. This taper design was used to demonstrate a tunable hybrid silicon laser directly modulated at 10 Gb/s [324]. An alternative coupler design concept, which can be accomplished solely with topdown fabrication of 3D tapered waveguides, is illustrated in Figure 11.21 as Design III. This coupler is adiabatically tapered both along its length and its width, which is expected to minimize scattering losses at the tips/edges of the tapered waveguide and thus improve coupling efficiency. The spatial mode distributions for the different cross sections of Design III are shown in Figure 11.21(g)−(i). A mode profile for a thin (40 nm) bonding layer is shown in Figure 11.21(i) and for a thick (325 nm) bonding layer in Figure 11.21(j). Note that the mode pulling into the silicon layer can take place over a relatively large range of bonding layer thicknesses. Figure 11.22 shows the 3D finite-difference time-domain (FDTD) simulation of the 3D taper. All dimensions are identical to Design III in Figure 11.21 and the taper length is 15 μm. In the simulation, a light pulse is launched inside the gain media and propagates toward the right side. After traveling through the 3D taper, the pulse is gradually pulled into the silicon waveguide. We have extracted and calculated the coupling efficiencies for the coupling between the gain media and the silicon waveguide. Note that the coupling efficiency is defined as the ratio between the output power into the Si waveguide and the power injected into the gain media. In conclusion, tuning several structural parameters, namely, bonding layer width, silicon waveguide cross-section dimensions, and III-V waveguide width, allows one to design a high-performance hybrid device for a specific application. Note that there is no preferential choice between Designs I and II, since their properties are quite similar in terms of threshold current, power consumption, and coupling efficiency [317, 318, 322], while Design III remains to be experimentally tested.

11.5.4

Perspective: Pushing the Footprint of DFBs to the Nanoscale While narrow linewidth and frequency stability are arguably the most important attributes of DFB lasers, the footprint is also an important characteristic. Future dense chip-scale

264

Application of Nanolasers: Photonic Integrated Circuits and Other Applications

integration of DFBs with photonic integrated circuits will benefit from size reduction. In the following perspective, we introduce concepts from nanoscale resonant devices to assess the possibility of DFB lasers with extremely small footprints. For simplicity, we assume that bonding of such lasers may be achieved using techniques discussed in earlier sections. The cavity Q characterizes the photonic “transit” time, or duration that a photon will stay in the cavity. Generally, Q factor is expressed in terms of cavity loss channels, Q−1 = Q−1rad + Q−1dissip + Q−1other, including radiative, dissipative, and all others denoted by Qrad, Qdissip, and Qother, respectively [302]. To achieve a narrow linewidth, Q must be very large. For small cavities, with dimensions on the order, or even smaller, than the free-space wavelength, maintaining a large Qrad is feasible only with the introduction of metallic constituents. However, the metal reduces Qdissip, which indicates an inherent trade-off. In Chapter 2, we saw that the introduction of a low-index dielectric “shield” layer between the metal-cladding and gain region can greatly increase Qdissip, while still reaping the benefits of a high Qrad [99]. Beyond the metallo-dielectric configuration, coaxial metal geometries can be used. Coaxial optical resonators, like their RF counterparts, support modes to arbitrarily small spatial dimensions, ultimately limited only by fabrication technologies. As we saw in Chapter 4, Section 4.5, subwavelength nanoscale coaxial lightemitting devices can exhibit intriguing physical behavior, for example, the so-called thresholdless light-light curve [14]. While true thresholdless behavior remains to be verified from a photon statistics point of view, the efficiency of spontaneous emission coupling to the lasing mode is maximized in these coaxial resonators. State-of-the-art, subwavelength, semiconductor lasers are generally surface emitting and, therefore, are not easily integrated with waveguides and other on-chip devices. Although we discussed several approaches to increase the coupling in Section 11.3, experimental demonstration of such proposals remains to be seen. Since the late 2000, hyperbolic metamaterials (HMMs) have emerged as a new area in the metamaterial community. HMMs can potentially assist in the chip-level integration of lasers and other components, as HMMs consisting of III-V compounds offer a potential means of creating integrated, in-plane subwavelength semiconductor lasers. HMMs are periodic composite media with deeply subwavelength layers of metal and dielectric that exhibit unique physical properties [325]. In particular, HMMs support modes, in the ideal limit, with arbitrarily large spatial frequency. This attribute can be appreciated through the dispersion relation for HMMs. Assuming a uniaxial crystal, three dispersion relations are possible, depending on the signs of the elements of the diagonal permittivity tensor. For the following inequalities, εxx ; εyy ; εzz > 0

ð11:19Þ

εxx ; εyy < 0; εzz > 0

ð11:20Þ

εxx ; εyy > 0; εzz < 0

ð11:21Þ

11.5 Silicon-compatible Miniature Laser

265

the corresponding dispersion relations are 2 2 ω2 kx þ ky kz2 ¼ þ c2 εzz εxx

ð11:22Þ

2 2 ω 2 kx þ ky k2 ¼  z 2 c εzz jεxx j

ð11:23Þ

kx2 þ ky2 kz2 ω2 þ ¼  c2 jεzz j εxx

ð11:24Þ

Equations (11.19) and (11.22) describe a purely dielectric crystal, whereas Equations (11.20), (11.21), (11.23), and (11.24) describe a composite medium with metal and dielectric constituents. While Equation (11.22) describes an ellipsoid in reciprocal space, Equations (11.23) and (11.24) describe open and closed hyperboloids (e.g., see fig. 1 of reference [325]). The hyperboloids diverge for increasing spatial frequency, which mathematically describes the fact that deeply subwavelength field confinement is possible in HMMs. Physically, the transmission through HMMs is due to the strong coupling of SPPs at adjacent metal-dielectric interfaces [326]. Because of the high spatial frequency modes supported by HMMs, waveguides with extremely small cross sections can be envisioned. By substituting the passive dielectric constituent with a III-V semiconductor, one can introduce optical gain to offset dissipative losses caused by the presence of metal. For applications requiring telecommunication frequencies, either noble metals or transparent conducting oxides (TCO) can be used for the metallic constituent. In the limit of effective medium theory, the combination of silver and InGaAsP, under moderate pumping levels, can lead to complete loss compensation for a propagating waveguide mode [194]. A schematic of an in-plane HMM waveguide is shown in Figure 11.23(a). The waveguide height and width, h and w1, can be simultaneously deeply subwavelength. For example, the cylindrical silver/InGaAsP HMM waveguide of reference [194] supports lossless propagation of the fundamental TM01 mode in a radius range of 50 to 150 nm, for moderate pumping levels, and metal fill fractions between 30% and 70%. In the rectangular waveguide of Figure 11.23(a), the TM11 mode is the fundamental mode. Assuming a square geometry for simplicity, the waveguide dimensions of 50 nm < h = w1 < 200 nm enable lossless propagation in the quasi-static effective medium limit. The extension of a waveguide-integrated HMM exhibiting lossless transmission to one with lasing requires a mechanism to provide positive feedback. This might be achieved through a periodic modulation of the waveguide width as shown schematically in Figure 11.23(b). The modulation of the waveguide width has a period on the order of the emission wavelength, which is much greater than the deeply subwavelength periodicity of the HMM. As a result of the high-k modes supported by the HMM, the widths of the narrow and wide sections of the waveguide, w1 and w2, respectively, can both be less than 100 nm. This enables densely packed waveguides for WDM applications. A number of challenges exist concerning the realization of a nanoscale hybrid-DFB. These include deposition of metal into the nanostructured III-V compound, as well as the

266

Application of Nanolasers: Photonic Integrated Circuits and Other Applications

Figure 11.23

Schematic of in-plane InGaAsP/Metal HMM, (a) without and (b) with periodic modulation of the waveguide width. For the modulated case, one-and-a half wavelength-scale periods are shown. Inset shows electric field distribution (|E|) of the lowest-order TM mode in a Ag/InGaAsP square waveguide (w1 = h = 100 nm) on silicon in the effective medium limit. Reprinted from reference [312] with permission from Optical Society of America (OSA).

design of electrical contacts. If a noble metal is used as the metal, the thickness of the individual metal layers should be < 30 nm to enable coupling between adjacent layers. This pushes the limit of physical deposition techniques, such as electron-beam evaporation and sputtering [327]. On the other hand, if TCO is used for the metal, the individual metal layers can be significantly thicker due to the much larger skin depth of TCOs [328]. Highly conformal thin films of TCOs may be deposited via atomic layer deposition (ALD), a monolayer-at-a-time chemical technique originally developed for oxides [328]. Deposition of noble metals via ALD is also feasible; however, ALD of singleelement materials is inherently more difficult than ALD of compounds [329]. While an in-plane HMM based on noble metals may be challenging to realize in practice, we note that the proposed device concept is only one of many possibilities. In this section, we looked into ways to reduce DFB dimensions for greater density of integration that may be achieved by modulated active HMM waveguides. While the quality factor of such devices will suffer due to the presence of metal, the footprint will be greatly reduced compared to state-of-the-art DFB lasers.

11.6

Other Applications and Future Trends of Nanolasers Some applications have already benefited from the advantages offered by nanolasers, in particular their extremely localized and high-intensity fields. We have focused on applying nanolasers in various aspects of dense chip-scale integrated circuits, which is by far the most promising and fought after application for nanolasers. Nanolasers (and nano-scale emitters in general) can find various applications. For the more conventional applications such as communications, computing, and light sources, nanolasers still

11.6 Other Applications and Future Trends of Nanolasers

267

require substantial improvements in (i) room-temperature threshold current, which is currently around a milli-ampere but could in theory be much lower [330]; (ii) device lifetime; and (iii) efficient out-coupling of light or plasmons into either free space or a (plasmonic) waveguide structure. We have aimed at addressing these challenges throughout this book. By their nature, the optical output power of nanolasers is small, relative to their more macroscopic counterparts. To increase the total output power, nanolasers may be grouped together in arrays where the modes of individual devices are coupled to form a supermode. Such coupled arrays have been demonstrated with photonic crystal lasers [331], metallic bowtie lasers [91], nanowire lasers [220], liquid crystal lasers [332], and nanoparticles lasers [333] acting as the individual elements. In addition to higher powers, some coupled arrays may attain higher efficiencies relative to their standalone element efficiencies [331]. For applications in communications, sensing, and beam steering, however, uncoupled nanolaser arrays are desired. While the output power of the individual elements remains small compared to the total power of the entire array, individually addressable, decoupled elements are essential for creating independent channels for guided and freespace signal propagation [334]. Additionally, beam steering of a phased array necessitates individually addressable, uncoupled elements [335]. Because of their size, uncoupled nanolasers may be packed at enormous densities, compared to their macroscopic counterparts. For example, the die of a state-of-the-art commercially available 4×1 VCSEL array has a footprint of 1000 μm × 300 μm (0.3 mm2), with a pitch between lasers of 250 μm [336]. Comparatively, a recent photonic crystal array used for live-cell imaging has a pitch of 5 μm [334]. This means that more than 12,000 photonic crystal lasers may be placed in the same footprint of the four commercial VSCELs. This comparison is, however, unfair, because the mentioned PC lasers are optically pumped. The recently demonstrated nanophotonics phased array, on the other hand, is, like commercial VCSELs, electronically controlled. It contained more than 4,000 elements in roughly the same footprint (0.33 mm2) as the VCSELs [335]. Nanolasers with metal-cladding offer even greater packing density. The electrically pumped laser discussed in the previous section [36] had a diameter of 1.5 μm. If we assume a pitch of twice this length, then more than 30,000 electronically controlled, metal-clad nanolasers could be placed in the same footprint as four commercial VCSELs. Using the coaxial structure, it is conceivable that this number could be even greater. Such a densely packed array would be ideal for high-resolution imaging/ sensing. Depending on their Purcell factors, metal-clad nanolasers may exhibit relaxation bandwidths that make them suitable for receiving, transmitting, and processing information at the chip scale. The combined density, low-power consumption and potentially fast operation make metal-clad nanolaser arrays strong candidates for light sources of future integrated photonic circuits. Another significant and exciting area of recent progress has been the application of small lasers in biology and the implementation of small lasers in biological structures. Some of the early works, as we will see later, use micro-scale lasers at the current stage of research.

268

Application of Nanolasers: Photonic Integrated Circuits and Other Applications

Figure 11.24

(a) Small lasers are also useful for ultrasensitive detection, such as with a fiber-coupled whispering-gallery microlaser (top). Here, the binding of individual influenza A (InfA) virus particles leads to mode splitting and thus to a beat frequency that relates to the number of bound particles (down). (b) A single biological cell can form a small laser if genetically programmed to produce green fluorescent protein. Schematic of the first single-cell laser (top) and typical spatial emission pattern (down). Part (a) reprinted from reference [337] with permission from Macmillan Publishers Ltd; part (b) reprinted from reference [339] with permission from Macmillan Publishers Ltd.

Small, low-power, efficient laser sources are useful for sensing applications, particularly when coupled with integrated optical cavities. Plasmonic resonators provide a broad bandwidth with strong spontaneous emission enhancement owing to their small cavity volume, whereas dielectric resonators tend to be larger but offer narrower bandwidth. Small lasers have also been used as sensors directly. The high sensitivity of microdisk lasers to changes in the ambient refractive index, for example, has enabled the detection of single virus particles attaching to the surface of an erbium-doped silica microdisk laser (Figure 11.24(a)) [337]. Another study found that protein adsorption kinetics can be measured with improved signal-to-noise ratio when using whisperinggallery mode resonators based on fluorescent polymer microspheres and operating these

11.6 Other Applications and Future Trends of Nanolasers

269

above the laser threshold [338]. For biosensing, lasers based on biologically produced materials such as green fluorescent protein [339, 340] or fluorescent vitamins [341] are uniquely suited to generate laser light within living organisms (Figure 11.24(b)). Using the small laser configurations, intracellular lasing could be achieved without an external resonator, which may enable novel nonlinear imaging schemes or dense wavelength multiplexing. Microlasers based on fluorescently labeled DNA as a gain medium may find applications in optical DNA sequencing [342]. For lasers to flourish in bioresearch, different trade-offs will be required between the various performance and fabrication parameters than for optical communication or data processing. For instance, diagnostics applications may require large quantities of small lasers, thus necessitating extremely cost-efficient fabrication but putting less stringent requirements on long-term operational stability. The maximum size of lasers applied within biological structures is imposed by nature and will vary between applications; for example, mammalian cells measure 10–50 μm, capillary blood vessels are a few micrometers in diameter, and the endocytic vesicles responsible for the transport of molecules and small particles across cell membranes typically have submicrometer dimensions.

Appendix A Spontaneous Emission in Free Space and Cavity

A.1

Nonrelativistic QED in Free Space and in a Resonant Cavity Following the formalism of ([231], §III.A.1), we begin the nonrelativistic quantum electrodynamics (QED) description of the electric field in free space and in a cavity by separating the longitudinal and transverse components of the electric field operator, E^ ¼ E^∥ þ E^⊥ . The longitudinal field operator E^∥ is fully determined by the charge distribution and describes the quasi-static field of charged particles. In what follows, we model electron-hole pairs in the gain material as two-level quantum systems, and cavity materials with their macroscopic permittivities ɛ; the model includes no charged particles. We therefore focus on the source-free condition and ignore E^∥ . The transverse component of a free field is given by ([231], §III.B.2), sffiffiffiffiffiffiffiffiffiffiffi  ℏωk  † ikr ikr ^ ^ i a ðtÞe  a ðtÞe E^⊥ ðr; tÞ ¼ ε k;ε k;ε 2ε0 L3 k;ε X

ðA:1Þ

In Equation (A.1), the summation is over all free-space modes, k is the wavevector of the mode, and ε is the polarization unit vector of the mode, satisfying ε⊥k. ωk ¼ jkjc is the mode frequency, L3 is the quantization volume, ^a †k;ε ðtÞ and ^a k;ε ðtÞ are photon creation and annihilation operators for the mode, respectively, and ^ a k;ε ðtÞ ¼ ^ a k;ε ð0Þ  eiωk t ;

^a †k;ε ðtÞ ¼ ^a †k;ε ð0Þ  eiωk t

ðA:2Þ

where ^ a ð0Þ and ^ a † ð0Þ are the operator values at time t = 0. Equations (A.1) and (A.2) are written for a free field in the Heisenberg picture, in which quantum states are constant and operators vary with time. They also apply in the Dirac picture for a field interacting with, for example, a two-level emitter if the interaction is included as correction to the unperturbed Hamiltonian. In this situation, the quantum states evolve due to the interaction ([134], §5.5). It is often convenient to separate Equation (A.1) into annihilation þ  E^⊥ and creation E^⊥ terms,

Appendix A: Spontaneous Emission in Free Space and Cavity

271

þ  E^⊥ ðr; tÞ ¼ E^⊥ ðr; tÞ þ E^⊥ ðr; tÞ; sffiffiffiffiffiffiffiffiffiffiffi X ℏωk þ ^ E ⊥ ðr; tÞ ¼ i^ a k;ε ðtÞeikr ε ; 2ε0 L3 k;ε

sffiffiffiffiffiffiffiffiffiffiffi  þ † X ℏωk †  ^ ^ E ⊥ ðr; tÞ ¼ E ⊥ ðr; tÞ ¼  i ^a k;ε ðtÞeikr ε 3 2ε L 0 k;ε

ðA:3Þ

An analogous representation exists for the electric field operator in a cavity [343, 344]. In a source-free cavity, the electric field operator becomes  X pffiffiffiffiffiffiffiffi þ  i ℏωk ^a k ðtÞ  ^a †k ðtÞ  ek ðrÞ ðA:4Þ E^u ðr; tÞ ¼ E^ u ðr; tÞ þ E^ u ðr; tÞ ¼ ωk >0

where the summation is over all cavity modes and ωk is the eigenfrequency of the mode k. In Equation (A.4), r is the location at which the field is evaluated, ek ðrÞ is the electric field modal profile normalized so that the mode energy evaluates to   1 ℏωk ^ a †k ðtÞ þ ^ a †k ðtÞ^a k ðtÞ , that is, ħωk per quantum level of the harmonic oscila k ðtÞ^ 2 1 lator and ℏωk in the oscillator ground state. Explicitly, in non-dispersive media, 2

ð   Ek ðrÞ ek ðrÞ ¼ pffiffiffiffiffiffi ; Nk ≡ εðrÞE2k ðrÞ þ μðrÞH2k ðrÞ d 3 r Nk V

ðA:5Þ

where Nk is the normalization factor for mode k and the integration is over the entire volume in space. Ek ðrÞ and Hk ðrÞ represent real cavity mode fields (solutions of the classical Maxwell’s equations for the cavity geometry), and integration is over all space. In electrically dispersive but magnetically non-dispersive media, Nk becomes [54]  2  3 ð ∂ ω0 εR ðr; ω0 Þ 2 2 5 3 Nk ¼ 4 0 Ek ðrÞ þ μðrÞHk ðrÞ d r ∂ω0 V ω ¼ωk  20  1 3 ð ∂ ω0 εR ðr; ω0 Þ 2 ¼ 4@ þ εR ðr; ωk ÞAEk ðrÞ5d 3 r 0 0 ∂ω V ω ¼ωk ðA:6Þ where εR stands for the real part of permittivity ε. The assumed, non-dispersive magnetic permeability enables us to express the total magnetic energy in Equation (A.6) in terms of the electric field [344]. Although εR may be negative in some metallic materials, the integral in Equation (A.6) is always positive. Note that the preceding formalism lacks the imaginary part of the permittivity and therefore ignores damping in the cavity. Damping may be introduced using Heisenberg-Langevin reservoir theory ([129], §9). We discuss such an approach to damping in the rest of this section. When the electromagnetic mode interacts with the environment, the time dependence of ^ a k ðtÞ and ^ a †k ðtÞ can no longer be described by Equation (A.2). A damping

272

Appendix A: Spontaneous Emission in Free Space and Cavity

environment can often be modeled as a thermal reservoir. The reservoir model is applicable when the interaction is weak and the environment is a large stochastic system that satisfies the Markovian approximation, namely, a system that over a short time τreservoir becomes fully disordered and loses all memory of its earlier state. Intuitively, the interaction must be sufficiently weak and the reservoir characteristic time τreservoir sufficiently short, so the mode experiences all possible states of the reservoir in equal measure. We employ the reservoir formalism to describe loss at the boundary of the cavity. Hereafter the terms “environment” and “reservoir” are used interchangeably. When a mode interacts with a thermal reservoir, the evolution of the mode operators ^ a k ðtÞ and ^ a †k ðtÞ also becomes stochastic. As a result, only statistical correlations involving ^ a k ðtÞ and ^a †k ðtÞ can be predicted for each mode. The correlations obey [129] 1

 Ck jτj  d † 2 ^ a k ðt þ τÞR ¼ Ck ½^a †k ðtÞ^ a k ðt þ τÞ R þ Ck nðωk Þe eiωk τ a k ðtÞ^ dt

1

 2Ck jτj   d ^ a †k ðt þ τÞR ¼ Ck ½^a k ðtÞ^ a †k ðt þ τÞ R þ Ck nðωk Þ þ 1 e eiωk τ a k ðtÞ^ dt

ðA:7Þ

where ½. . .R denotes the statistical expected value, and nðωk Þ represents the reservoir energy at frequency ωk. In Equation (A.7), Ck is the mode-reservoir coupling constant; thus, 1/Ck represents the cavity damping time. The expected value ½^a †k ðtÞ^a k ðtÞR of the photon count decays exponentially with the damping constant 1/Ck toward its steadystate value nðωk Þ, which is usually referred to as the reservoir temperature. Comparing the reservoir characteristic time τreservoir with the cavity damping time, the modereservoir weak-coupling condition is τreservoir > 1/Ck, the evolution of the correlation, which is described by Equation (A.7), reaches steady state, with its behavior described by Equation (A.8): 1  Ck jτj  †  2 ^ a k ðtÞ^ eiωk τ a k ðt þ τÞ R ¼ nðωk Þe 1  Ck jτj     2 ^ eiωk τ a †k ðt þ τÞ R ¼ nðωk Þ þ 1 e a k ðtÞ^

ðA:8Þ

Once mode-reservoir equilibrium has been reached, the correlations on the left-hand side of Equation (A.8) are fully determined by Ck and nðωk Þ. We next introduce the interaction between the electromagnetic field and a two-level emitter, such as an electron-hole pair in a semiconductor laser. Suppose the emitter is prepared at time t = t0 in its upper state |2>. The emitter interacts with the electromagnetic field mode, and the two become quantum mechanically entangled. At some later time t > t0, a phase-destroying event occurs, for example, a collision between two electrons in the conduction band of semiconductors [133]. Such an event either makes the emitter transition to the lower state |1> while simultaneously adding a photon of

Appendix A: Spontaneous Emission in Free Space and Cavity

273

frequency ω21 to the field or leaves the emitter in the upper state |2> and the mode with its original photon count. The emitter-mode interaction then begins anew and continues until the next phase-destroying event occurs. When such events are much more frequent than level transitions (transitions between states), the photonemission probability between time t0 and a later time t > t0 is small and is given by [345] P2→1;i ðtÞ ¼

1 ℏ

ð t0 þτcoll ð t0 þτcoll ð

2

t0

00

t0 Þ

t0

   þ   ij ℘ 12 ðω21 Þ  E^⊥ ðr; t0 Þ  ℘12 ðω21 Þ  E^⊥ ðr; t00 Þ ji Dðω21 Þdω21 dt0 dt00

〈



eiω21 ðt

〉

ðA:9Þ where ji〉 is the initial state of the field, and ℘12 ðω21 Þ is the dipole matrix element. ℘12 ðω21 Þ is a property of the emitter and determines the potential strength of the emittermode interaction ([137], §4.3). The actual interaction strength depends on the orientation of the dipole relative to the electric field and is thus governed by the dot product between the two. D(ω21) is the density of emitter states, which characterizes the inhomogeneity of the system (Dðω21 Þ ¼ δðω21  ω 21 Þ if all emitters are identical with natural frequency ω 21 ). Equation (A.9) is valid over time intervals short enough such that P2→1(t) ; this is referred to as spontaneous emission. We apply Equation (A.9) in free space, with all free-space modes in the vacuum state and no reservoir present. The field operators in this case have deterministic time dependences described by Equation (A.2). By substituting Equations (A.1) and (A.2) into Equation (A.9), we recover the Weisskopf-Wigner probability of spontaneous emission in the limit of a two-level system when D(ω21) = δ(ω–ω21) [349],

274

Appendix A: Spontaneous Emission in Free Space and Cavity

free P2→1;j0:::0〉 ¼

1

ð t0 þτcoll ð t0 þτcoll ð

ℏ2 t0 0

t0

eiω21 ðt

00

0

t Þ




1 sffiffiffiffiffiffiffiffiffiffiffi X ℏωk00 00 † 00 ik00 re A ^ ε ðt Þe a @ ℘12 ðω21 Þ  0 . . . 0 Dðω21 Þdω21 dt0 dt00 k00 ;ε00 2ε0 L3 k00 ;ε00 2 X ωk ð ℘ ðω21 Þ  ε Dðω21 ÞRðωk  ω21 ; τcoll Þdω21 ¼ 12 3 2ℏε L k;ε

ð ≈

0

ω321 τcoll j ℘12 ðω21 Þj2 Dðω21 Þdω21 3πℏε0 c3

ðA:10Þ

In Equation (A.10), re is the location of the emitter, and summation cross-terms 〈0 . . . 0j^ a k0 ;ε0 ^ a †k00 ;ε00 j0 . . . 0〉 ¼ δk0 k00 δε0 ε00 . The quantity i h1 ð t0 þτcoll ð t0 þτcoll sin ðω  ω21 Þτcoll 2 00 0 2 Rðω  ω21 ; τcoll Þ≡ eiðωω21 Þðt t Þ dt0 dt00 ¼ , 1 t0 t0 ðω  ω21 Þ

cancel

owing

to

2

which absorbs the time exponents inserted from Equation (A.2), is the homogeneous broadening function and depends on τcoll. Viewed as a function of ω, R(ω–ω21,τcoll) peaks at ω21, has a width on the order of 1/τcoll, and satisfies ð Rðω  ω21 ; τcoll Þdω ¼ 2π  τcoll [134]. The approximation in Equation (A.10) consists in replacing the summation over free-space modes k with appropriate integration and then taking ωk ≈ ω21. Such an approximation is justified because the -free-space modes form a continuum with an infinitesimal spectral spacing between adjacent modes, and the quantity ω3k varies little over the width of R(ω–ω21,τcoll). A similar procedure can be carried out in an undamped cavity if all cavity modes are initially in a vacuum state. Applying Equation (A.4) to Equation (A.9), summation cross-terms cancel again according to 〈0 . . . 0j^a k0 ^a †k 00 j0 . . . 0〉 ¼ δk 0 k 00 , and we obtain X ωk ð cav P2→1;j0:::0〉 ¼ ðA:11Þ j ℘12 ðω21 Þ  ek ðre Þj2 Dðω21 ÞRðωk  ω21 ; τcoll Þdω21 ℏ k Unlike in free space, the summation over modes k in Equation (A.11) cannot be replaced with integration if the spectral spacing between adjacent modes is non-negligible. This is especially the case in micro- and nanocavities in which the spacing between adjacent modes may be a significant fraction of the modes’ resonance frequencies. The cavity spontaneous emission probability given by Equation (A.11) may depend significantly on the number of available modes and their location relative to the density of emitter states D(ω21). It also depends on the location and orientation of the emitter relative to the normalized mode field ek ðrÞ. For example, the probability is zero for an emitter located at a field node.

Appendix A Spontaneous Emission in Free Space and Cavity

A.1

Nonrelativistic QED in Free Space and in a Resonant Cavity Following the formalism of ([231], §III.A.1), we begin the nonrelativistic quantum electrodynamics (QED) description of the electric field in free space and in a cavity by separating the longitudinal and transverse components of the electric field operator, E^ ¼ E^∥ þ E^⊥ . The longitudinal field operator E^∥ is fully determined by the charge distribution and describes the quasi-static field of charged particles. In what follows, we model electron-hole pairs in the gain material as two-level quantum systems, and cavity materials with their macroscopic permittivities ɛ; the model includes no charged particles. We therefore focus on the source-free condition and ignore E^∥ . The transverse component of a free field is given by ([231], §III.B.2), sffiffiffiffiffiffiffiffiffiffiffi  ℏωk  † ikr ikr ^ ^ i a ðtÞe  a ðtÞe E^⊥ ðr; tÞ ¼ ε k;ε k;ε 2ε0 L3 k;ε X

ðA:1Þ

In Equation (A.1), the summation is over all free-space modes, k is the wavevector of the mode, and ε is the polarization unit vector of the mode, satisfying ε⊥k. ωk ¼ jkjc is the mode frequency, L3 is the quantization volume, ^a †k;ε ðtÞ and ^a k;ε ðtÞ are photon creation and annihilation operators for the mode, respectively, and ^ a k;ε ðtÞ ¼ ^ a k;ε ð0Þ  eiωk t ;

^a †k;ε ðtÞ ¼ ^a †k;ε ð0Þ  eiωk t

ðA:2Þ

where ^ a ð0Þ and ^ a † ð0Þ are the operator values at time t = 0. Equations (A.1) and (A.2) are written for a free field in the Heisenberg picture, in which quantum states are constant and operators vary with time. They also apply in the Dirac picture for a field interacting with, for example, a two-level emitter if the interaction is included as correction to the unperturbed Hamiltonian. In this situation, the quantum states evolve due to the interaction ([134], §5.5). It is often convenient to separate Equation (A.1) into annihilation þ  E^⊥ and creation E^⊥ terms,

Appendix A: Spontaneous Emission in Free Space and Cavity

271

þ  E^⊥ ðr; tÞ ¼ E^⊥ ðr; tÞ þ E^⊥ ðr; tÞ; sffiffiffiffiffiffiffiffiffiffiffi X ℏωk þ ^ E ⊥ ðr; tÞ ¼ i^ a k;ε ðtÞeikr ε ; 2ε0 L3 k;ε

sffiffiffiffiffiffiffiffiffiffiffi  þ † X ℏωk †  ^ ^ E ⊥ ðr; tÞ ¼ E ⊥ ðr; tÞ ¼  i ^a k;ε ðtÞeikr ε 3 2ε L 0 k;ε

ðA:3Þ

An analogous representation exists for the electric field operator in a cavity [343, 344]. In a source-free cavity, the electric field operator becomes  X pffiffiffiffiffiffiffiffi þ  i ℏωk ^a k ðtÞ  ^a †k ðtÞ  ek ðrÞ ðA:4Þ E^u ðr; tÞ ¼ E^ u ðr; tÞ þ E^ u ðr; tÞ ¼ ωk >0

where the summation is over all cavity modes and ωk is the eigenfrequency of the mode k. In Equation (A.4), r is the location at which the field is evaluated, ek ðrÞ is the electric field modal profile normalized so that the mode energy evaluates to   1 ℏωk ^ a †k ðtÞ þ ^ a †k ðtÞ^a k ðtÞ , that is, ħωk per quantum level of the harmonic oscila k ðtÞ^ 2 1 lator and ℏωk in the oscillator ground state. Explicitly, in non-dispersive media, 2

ð   Ek ðrÞ ek ðrÞ ¼ pffiffiffiffiffiffi ; Nk ≡ εðrÞE2k ðrÞ þ μðrÞH2k ðrÞ d 3 r Nk V

ðA:5Þ

where Nk is the normalization factor for mode k and the integration is over the entire volume in space. Ek ðrÞ and Hk ðrÞ represent real cavity mode fields (solutions of the classical Maxwell’s equations for the cavity geometry), and integration is over all space. In electrically dispersive but magnetically non-dispersive media, Nk becomes [54]  2  3 ð ∂ ω0 εR ðr; ω0 Þ 2 2 5 3 Nk ¼ 4 0 Ek ðrÞ þ μðrÞHk ðrÞ d r ∂ω0 V ω ¼ωk  20  1 3 ð ∂ ω0 εR ðr; ω0 Þ 2 ¼ 4@ þ εR ðr; ωk ÞAEk ðrÞ5d 3 r 0 0 ∂ω V ω ¼ωk ðA:6Þ where εR stands for the real part of permittivity ε. The assumed, non-dispersive magnetic permeability enables us to express the total magnetic energy in Equation (A.6) in terms of the electric field [344]. Although εR may be negative in some metallic materials, the integral in Equation (A.6) is always positive. Note that the preceding formalism lacks the imaginary part of the permittivity and therefore ignores damping in the cavity. Damping may be introduced using Heisenberg-Langevin reservoir theory ([129], §9). We discuss such an approach to damping in the rest of this section. When the electromagnetic mode interacts with the environment, the time dependence of ^ a k ðtÞ and ^ a †k ðtÞ can no longer be described by Equation (A.2). A damping

272

Appendix A: Spontaneous Emission in Free Space and Cavity

environment can often be modeled as a thermal reservoir. The reservoir model is applicable when the interaction is weak and the environment is a large stochastic system that satisfies the Markovian approximation, namely, a system that over a short time τreservoir becomes fully disordered and loses all memory of its earlier state. Intuitively, the interaction must be sufficiently weak and the reservoir characteristic time τreservoir sufficiently short, so the mode experiences all possible states of the reservoir in equal measure. We employ the reservoir formalism to describe loss at the boundary of the cavity. Hereafter the terms “environment” and “reservoir” are used interchangeably. When a mode interacts with a thermal reservoir, the evolution of the mode operators ^ a k ðtÞ and ^ a †k ðtÞ also becomes stochastic. As a result, only statistical correlations involving ^ a k ðtÞ and ^a †k ðtÞ can be predicted for each mode. The correlations obey [129] 1

 Ck jτj  d † 2 ^ a k ðt þ τÞR ¼ Ck ½^a †k ðtÞ^ a k ðt þ τÞ R þ Ck nðωk Þe eiωk τ a k ðtÞ^ dt

1

 2Ck jτj   d ^ a †k ðt þ τÞR ¼ Ck ½^a k ðtÞ^ a †k ðt þ τÞ R þ Ck nðωk Þ þ 1 e eiωk τ a k ðtÞ^ dt

ðA:7Þ

where ½. . .R denotes the statistical expected value, and nðωk Þ represents the reservoir energy at frequency ωk. In Equation (A.7), Ck is the mode-reservoir coupling constant; thus, 1/Ck represents the cavity damping time. The expected value ½^a †k ðtÞ^a k ðtÞR of the photon count decays exponentially with the damping constant 1/Ck toward its steadystate value nðωk Þ, which is usually referred to as the reservoir temperature. Comparing the reservoir characteristic time τreservoir with the cavity damping time, the modereservoir weak-coupling condition is τreservoir > 1/Ck, the evolution of the correlation, which is described by Equation (A.7), reaches steady state, with its behavior described by Equation (A.8): 1  Ck jτj  †  2 ^ a k ðtÞ^ eiωk τ a k ðt þ τÞ R ¼ nðωk Þe 1  Ck jτj     2 ^ eiωk τ a †k ðt þ τÞ R ¼ nðωk Þ þ 1 e a k ðtÞ^

ðA:8Þ

Once mode-reservoir equilibrium has been reached, the correlations on the left-hand side of Equation (A.8) are fully determined by Ck and nðωk Þ. We next introduce the interaction between the electromagnetic field and a two-level emitter, such as an electron-hole pair in a semiconductor laser. Suppose the emitter is prepared at time t = t0 in its upper state |2>. The emitter interacts with the electromagnetic field mode, and the two become quantum mechanically entangled. At some later time t > t0, a phase-destroying event occurs, for example, a collision between two electrons in the conduction band of semiconductors [133]. Such an event either makes the emitter transition to the lower state |1> while simultaneously adding a photon of

Appendix A: Spontaneous Emission in Free Space and Cavity

273

frequency ω21 to the field or leaves the emitter in the upper state |2> and the mode with its original photon count. The emitter-mode interaction then begins anew and continues until the next phase-destroying event occurs. When such events are much more frequent than level transitions (transitions between states), the photonemission probability between time t0 and a later time t > t0 is small and is given by [345] P2→1;i ðtÞ ¼

1 ℏ

ð t0 þτcoll ð t0 þτcoll ð

2

t0

00

t0 Þ

t0

   þ   ij ℘ 12 ðω21 Þ  E^⊥ ðr; t0 Þ  ℘12 ðω21 Þ  E^⊥ ðr; t00 Þ ji Dðω21 Þdω21 dt0 dt00

〈



eiω21 ðt

〉

ðA:9Þ where ji〉 is the initial state of the field, and ℘12 ðω21 Þ is the dipole matrix element. ℘12 ðω21 Þ is a property of the emitter and determines the potential strength of the emittermode interaction ([137], §4.3). The actual interaction strength depends on the orientation of the dipole relative to the electric field and is thus governed by the dot product between the two. D(ω21) is the density of emitter states, which characterizes the inhomogeneity of the system (Dðω21 Þ ¼ δðω21  ω 21 Þ if all emitters are identical with natural frequency ω 21 ). Equation (A.9) is valid over time intervals short enough such that P2→1(t) ; this is referred to as spontaneous emission. We apply Equation (A.9) in free space, with all free-space modes in the vacuum state and no reservoir present. The field operators in this case have deterministic time dependences described by Equation (A.2). By substituting Equations (A.1) and (A.2) into Equation (A.9), we recover the Weisskopf-Wigner probability of spontaneous emission in the limit of a two-level system when D(ω21) = δ(ω–ω21) [349],

274

Appendix A: Spontaneous Emission in Free Space and Cavity

free P2→1;j0:::0〉 ¼

1

ð t0 þτcoll ð t0 þτcoll ð

ℏ2 t0 0

t0

eiω21 ðt

00

0

t Þ




1 sffiffiffiffiffiffiffiffiffiffiffi X ℏωk00 00 † 00 ik00 re A ^ ε ðt Þe a @ ℘12 ðω21 Þ  0 . . . 0 Dðω21 Þdω21 dt0 dt00 k00 ;ε00 2ε0 L3 k00 ;ε00 2 X ωk ð ℘ ðω21 Þ  ε Dðω21 ÞRðωk  ω21 ; τcoll Þdω21 ¼ 12 3 2ℏε L k;ε

ð ≈

0

ω321 τcoll j ℘12 ðω21 Þj2 Dðω21 Þdω21 3πℏε0 c3

ðA:10Þ

In Equation (A.10), re is the location of the emitter, and summation cross-terms 〈0 . . . 0j^ a k0 ;ε0 ^ a †k00 ;ε00 j0 . . . 0〉 ¼ δk0 k00 δε0 ε00 . The quantity i h1 ð t0 þτcoll ð t0 þτcoll sin ðω  ω21 Þτcoll 2 00 0 2 Rðω  ω21 ; τcoll Þ≡ eiðωω21 Þðt t Þ dt0 dt00 ¼ , 1 t0 t0 ðω  ω21 Þ

cancel

owing

to

2

which absorbs the time exponents inserted from Equation (A.2), is the homogeneous broadening function and depends on τcoll. Viewed as a function of ω, R(ω–ω21,τcoll) peaks at ω21, has a width on the order of 1/τcoll, and satisfies ð Rðω  ω21 ; τcoll Þdω ¼ 2π  τcoll [134]. The approximation in Equation (A.10) consists in replacing the summation over free-space modes k with appropriate integration and then taking ωk ≈ ω21. Such an approximation is justified because the -free-space modes form a continuum with an infinitesimal spectral spacing between adjacent modes, and the quantity ω3k varies little over the width of R(ω–ω21,τcoll). A similar procedure can be carried out in an undamped cavity if all cavity modes are initially in a vacuum state. Applying Equation (A.4) to Equation (A.9), summation cross-terms cancel again according to 〈0 . . . 0j^a k0 ^a †k 00 j0 . . . 0〉 ¼ δk 0 k 00 , and we obtain X ωk ð cav P2→1;j0:::0〉 ¼ ðA:11Þ j ℘12 ðω21 Þ  ek ðre Þj2 Dðω21 ÞRðωk  ω21 ; τcoll Þdω21 ℏ k Unlike in free space, the summation over modes k in Equation (A.11) cannot be replaced with integration if the spectral spacing between adjacent modes is non-negligible. This is especially the case in micro- and nanocavities in which the spacing between adjacent modes may be a significant fraction of the modes’ resonance frequencies. The cavity spontaneous emission probability given by Equation (A.11) may depend significantly on the number of available modes and their location relative to the density of emitter states D(ω21). It also depends on the location and orientation of the emitter relative to the normalized mode field ek ðrÞ. For example, the probability is zero for an emitter located at a field node.

Appendix B Temperature-dependent Material Gain

B.1

Analysis of the Temperature-dependent Material Gain Spectrum of Bulk In0.53Ga0.47As The material emission spectra − the spontaneous emission and optical gain in the absence of a cavity − may be determined numerically with good accuracy following the semiclassical approach of [137]. The model of [137] treats the electric field classically and accounts for the quantum nature of the material system through the transition matrix element, MT, under the electric-dipole approximation [134]. To understand the behavior of a laser with respect to different temperatures, we need to understand the temperature dependence of the material system that provides the gain in the device. InGaAsP MQW and In0.53Ga0.47As bulk gain are used in numerous examples in this book, because they are the most commonly used semiconductor gain materials for nearIR telecommunication wavelengths. In this appendix, we analyze the temperature dependence of In0.53Ga0.47As bulk and InGaAsP MQW’s material gain spectrum. The key to a gain material is its ability to become “inverted” under the influence of an external pumping mechanism. By inversion, we mean that under certain situations, the system moves far from thermal equilibrium, and the charge carriers have a greater probability of being in an excited state than in the ground state. The inversion factor, f2 − f1, quantifies the degree to which a semiconductor is inverted. If the probability that an electron will occupy the conduction band, f2, exceeds the probability of it occupying the valence band, f1, then f2 − f1 > 0, and we have an inverted material that may amplify an electromagnetic field whose frequency corresponds to the transition frequency at which the inversion occurs. The inversion factor is a function of the pumping, which can be presented by the carrier density in the system. The amount of thermally excited carriers Nth contributing to the total carrier density N obviously increases with temperature. Therefore, if we constrain the total carrier density N to a fixed value and lower the temperature, we know that the contribution to N by Nth is decreasing. Therefore, the pumping agent must account for the balance of carriers, which means that the inversion factor must rise as we lower the temperature. This is observed in Figure B.1, where N is fixed at 1e18 cm−3 and the inversion factor is plotted for T of 77 K, 150 K, and 300 K over a wavelength range of 1200 nm to 1800 nm.

276

Appendix B: Temperature-dependent Material Gain

Figure B.1

Inversion factor versus wavelength for bulk InGaAs with a fixed carrier density of 1e18 cm−3 and T = 77, 150, and 300 K.

Conversely, if we were to fix the temperature, we would be holding Nth constant. If the carrier density is increased, the pumping agent is completely responsible for the increase, and the inversion factor will increase. Therefore, one should bear in mind that the inversion factor, carrier density, and temperature form a set of quantities such that if two are defined, the third one is completely determined. Furthermore, note that we have not yet said anything about the temperature or carrier-dependent bandgap. A consequence of the relationship between f2 − f1, N, and T is that the peak gain, being directly proportional to the inversion factor, is greater at the lower temperature for any carrier density. We can observe this behavior in Figure B.2, where the peak gain is plotted as a function of carrier density for T of 77 K, 150 K, and 300 K. In addition to the interdependence of f2 − f1, N, and T, the bandgap energy of the material depends strongly on the temperature and, to a lesser degree, on the carrier density. Both of these effects are usually formulated in terms of empirical relationships. For example, the bandgap energy as a function of temperature is expressed by Eg ðTÞ ¼ Eg ð0Þ 

αT 2 ðξ þ TÞ

ðB:1Þ

while its dependence on carrier density is expressed by Eg ðNÞ ¼ Eg  cN 1=3

ðB:2Þ

where α, ξ, and c are constants that fit to the empirical data. For bulk In0.53Ga0.47As Eg(0) = 0.818 eV, α = 4.09x10−4 eV/K, ξ = 224 K, and c = 20 meV/1018 cm−3 [137].

Appendix B: Temperature-dependent Material Gain

Figure B.2

Figure B.3

277

Inversion factor versus wavelength for bulk InGaAs with a fixed carrier density of 1e18 cm−3 and T = 77, 150, and 300 K.

Material gain spectrum of bulk In.53 Ga.47As for the temperature of T = 77 K, with carrier density of N = 1×1018, 2×1018, and 3×1018 cm−3. The effects of the carrier-dependent bandgap and band filling are visible.

In Figure B.3, we fix the temperature to T = 77 K and plot the gain spectrum for N = 1×1018, 2×1018, and 3×1018 cm−3. Now we see the less dramatic red-shift of the bandgap due to increasing carrier density, as suggested by Equation (B.2). Figure B.3 also shows that the maximum gain value with respect to the wavelength blue-shifts as the

278

Appendix B: Temperature-dependent Material Gain

Figure B.4

Material gain spectrum of bulk In.53 Ga.47As for the carrier density of N = 1×1018 cm−3, with T = 77 K, 150 K, and 300 K. The effect of the temperature-dependent bandgap is pronounced.

carrier density increases. Furthermore, we observe that the transparency point also blue-shifts. These effects are a consequence of band filling. As a result of the Pauli Exclusion Principle, as the carrier density increases, the carriers must occupy increasingly higher energy bands, pushing the center of mass of the spectrum to shorter wavelengths. In Figure B.4, we fix the carrier density to N = 1×1018 cm−3 and show the gain spectrum for T = 77, 150, and 300 K. Notice that the three curves cross the transparency point of g = 0 cm−1 at the identical wavelength as the inversion factor of Figure B.1. Additionally, we can observe the significant red-shift in the bandgap energy as the temperature increases, as suggested by Equation. (B.1). Throughout this analysis we have ignored the homogeneous broadening factor to isolate the effects so far presented. The two main results of including this term will be to smooth the discontinuity at the bandgap energy, thus enabling below-bandgap transitions, and to reduce the gain values by a modest amount, that is, no more than 10% for temperatures below 400 K. The model assumes a parabolic band structure and considers only the conduction band to heavy-hole valence band transition. The material parameters that we have used for bulk In0.53Ga0.47As include the conduction and valence band effective masses of mc = 0.046m0 and mv = 0.36m0 where m0 = 9.1×10−31 kg and the matrix transition

 1 25:3 2 m0 kg  eV[137]. element of jMT j ¼ 3 2

Appendix B: Temperature-dependent Material Gain

B.2

279

Analysis of the Temperature-dependent Material Gain spectrum of MQW InGaAsP Although data for the temperature dependence of MT in InGaAsP is lacking, MT in GaN was shown to be essentially constant with T in the range 300 K < T < 450 K in [350]. We therefore assume without further justification that MT is constant over the range of our interest, 75 K < T < 400 K, in 1.6Q InGaAsP, and use 2|M|2/m0 = 24.9 eV [351], where m0 is the free electron mass. At the bandedge of the conduction to heavy-hole (C-HH) interband transition, |MT|2 = |M|2/2 for TE polarization and |MT|2 = 0 for TM polarization. On the other hand, for the conduction to light-hole (C-LH) interband transition |MT|2 = |M|2/6 for TE polarization and |MT|2 = 2|M|2/3 for TM polarization, again at the bandedge. Here TE and TM polarizations refer to optical modes whose electric field vectors are perpendicular and parallel to the MQW growth direction, respectively. The calculation can be done following references [156, 157], approximating the band structure of InGaAsP with parabolic conduction and valence bands, characterized by the temperature-independent effective masses mC = 0.045m0 and mV = 0.37m0, respectively [351]. The temperature independence is justified on account that over the range 0 K < T < 300 K, mC only varies by less than 10% in GaAs and InP [352]. The effective masses determine the reduced density of states (DOS) function, which for a QW system is ρR ðE; TÞ ¼

 1 X mR  Θ E  ðEG þ EnC þ EnV Þ 2 LW n πℏ

ðB:3Þ

where mR = mCmV / (mC+mV) is the reduced effective mass, Θ(·) is the Heaviside step function, EG is the bandgap energy, and En,C and En,V are the conduction and valence subband energy levels, respectively. In the calculation of the subband energy levels, we assume k-selection rules and take the conduction and valence band offsets as 0.39ΔEG(T) and 0.61ΔEG(T), respectively, where ΔEG(T) = EG,1.3Q(T) − EG,1.6Q(T) [353]. For the 1.6Q/1.3Q InGaAsP QW, only one pair of subbands (E1 C, E1 V) exists due to the shallowness of the well. The emission spectra vary strongly with temperature and carrier density, to a large part, because of the temperature- and carrier-dependent bandgap energy, EG(N;T). Using Varshni’s empirical relation [354] and a phenomenological constant [137, 148] for the temperature and carrier dependences, respectively, the bandgap energy is expressed as EG ðN; TÞ ¼ EG ð0KÞ 

aT 2  cS N 1=3 bþT

ðB:4Þ

where EG,1.6Q(0 K) = 0.848 eV [137] and b = 224 K [355], and cS = 20 meV/(1012 cm−2/ LA)1/3 [137]. To find a, we fit Equation (B.4) to the documented EG,1.6Q(300 K) [16] and obtain a = 4.26×10−4 eV/K. The energy of the emitted photon E21 then is E21(T) = EG(T) + En,C(T) + En,V(T).

280

Appendix B: Temperature-dependent Material Gain

The temperature and carrier density are related through the Fermi distributions f2 and f1, describing the probability that an electron will occupy the conduction or valence band, respectively. For the active material to provide gain, the inversion factor, f2 − f1, must be positive. If the temperature and carrier density are both specified, then the bandgap energy is completely specified, and the inversion factor is determined through the relations [211] N ¼ NQW þ NUB NQW

NUB

4πmC X ¼ L W h2 n

ΔE ðC

dE 1 þ exp½ðE  FC Þ=ðkB TÞ

ðB:5Þ ðB:6Þ

EnC

pffiffiffiffi

 ð∞ EdE 4 2πmC 3=2 ¼ pffiffiffi 1 þ exp½ðE  FC Þ=ðkB TÞ h2 π

ðB:7Þ

ΔEC

Equation (B.5) expresses the given electron density as the sum of electrons confined to the QW, NQW, and unbound electrons propagating above the barrier layers, NUB. The integral of Equation (B.6) extends from the bottom of the first conduction subband to the extent of the barrier, ΔEC. Because only one pair of bound states exists, the summation truncates at n = 1. In Equations (B.6) and (B.7), FC is the quasi-Fermi level in the conduction band and kB is Boltzmann’s constant. Given N and T, Equations (B.6) and (B.7) can be solved for FC. Similarly, for the quasi-Fermi level in the valence band FV, we solve Equations (B.6) and (B.7), substituting P, mV, EnV, and ΔEV appropriately, where P is the hole density and P = N is assumed. When FC − FV > E21, f2 – f1 > 0, and the medium provides gain at E21, N, and T. The optical gain as a function of transition frequency, in dimensions of inverse length, is [137, 211] ð πq2 gðℏω; N; TÞ ¼ jMT ðE21 Þj2 ρR ðE21 ; TÞ½f2 ðN; TÞ nr ε0 cm0 ω f1 ðN; TÞRðℏω  E21 ; TÞdE21

ðB:8Þ

In Equation (B.8), q is the fundamental charge and R is the homogeneous broadening lineshape. For its temperature dependence, we use the theoretical results of [133], where the intraband lifetime was calculated as a function of temperature for In0.53Ga0.47As/InP QW at a carrier sheet density of 1×1012 cm−2. For the frequency dependence of nr, we use a function composed of the Sellmeir equations for the binaries of which 1.6Q InGaAsP is composed. Common material parameters for 1.6Q/1.3Q InGaAsP QW are listed in Table B.1. Figure B.5 shows the gain over the wavelength range of 1.1 μm < λ < 1.8 μm at T = 300 K with N as a parameter. The effect of the bandgap shrinkage with N is visible, as is the blue-shifting of the transparent wavelength due to band filling.

Appendix B: Temperature-dependent Material Gain

Table B.1 Common material parameters for 1.6Q/1.3Q InGaAsP QW. Symbol

Quantity

Value

RC-HH

Dipole moment for conduction to heavy- 1.22 nm hole interband transition at λ = 1.55μm

Reference [351]

EG,W(0 K)

Bandgap energy of well at 0 K

0.848 eV

[137]

EG,B(0 K)

Bandgap energy of barrier at 0 K

1.029 eV

[137]

EG,W(300 K)

Bandgap energy of well at 300 K

0.775 eV

[137]

EG,B(300 K)

Bandgap energy of barrier at 300 K

0.954 eV

[137]

a

Varshni parameter 1

4.26e-4 eV K

b

Varshni parameter 2

224 K

−1

Calculated [355]

12

−2

1/3

cS

Carrier-dependent bandgap constant

20 meV/(10

ΔEC/ΔEG,W

Conduction band (CB) offset

0.39

[127, 353]

ΔEV/ΔEG,W

Valence band (VB) offset

0.61

[127, 353]

mC/m0

CB effective mass

0.045

[351]

mC/m0

VB effective mass

0.37

[351]

EC,1

First CB subband energy (wrt to CB edge) 29.7 meV

Calculated

EV,1

First CB subband energy (wrt to VB edge) −3.6 meV

Calculated

τ(100 K)

Intraband lifetime at 75 K

0.93 ps

[133]

τ(400 K)

Intraband lifetime at 400 K

0.25 ps

[133]

Figure B.5

cm /10 nm)

Gain spectrum at T = 300 K with carrier density N as a parameter.

[17]

281

282

Appendix B: Temperature-dependent Material Gain

Figure B.6

Gain spectrum at N = 2.5e18 cm−3 with temperature as a parameter.

Figure B.6 shows gain spectrum over the same wavelength range, maintain N = 2.5×1018 cm−3, and vary T. The red-shifting of the peak gain with temperature is visible, as is the decrease in peak gain with temperature. The latter effect is explained if we consider the total carrier density N to be a sum of thermally excited carriers, Ntherm, and externally injected carriers Next. For a fixed N, Ntherm increases with temperature, meaning that Next decreases with T. The decreasing Next with T translates into decreasing FC − FV and f2 − f1, and thus lower peak gain.

Appendix C Modeling Thermal Effects in Nanolasers

C.1

Thermal Model Overview The thermal modeling strategy detailed here is based on models used in vertical-cavity surface-emitting lasers (VCSELs) [239], with modification to include features specific to nanoscale lasers. In the first step, various self-heating sources are detailed in Sections C.2−C.6, then the total self-heating power generated by the laser at a given operating current and operating temperature is calculated. These self-heating sources are located within the laser’s semiconductor layers, as well as at the semiconductor junctions. In the second step, described in Section 8.1, finite element COMSOL thermal simulation is used to model the temperature distribution throughout the laser and surrounding substrate/cladding, for the self-heating sources listed earlier. In the example used, the temperature increases are moderate, so the steady-state operating temperature is heated given the self-heating sources and material thermal parameters at the laser’s initial temperature. For a more complete analysis, necessary for a device with significant temperature changes, this process should be performed iteratively, with the self-heating sources and operating temperature recalculated at each time step. The designed room-temperature nanolaser in Section 8.5 is used as an example device in this appendix. The nanolaser with its accompanying electrical contacts are redrawn in Figure C.1. The radii and thicknesses of the other layers comprising the InP pedestals and InGaAsP top and bottom contact layers are given in Table C.1. The laser’s upper and lower InP pedestals both have radii less than rcore with an average ratio of rupper/rlower = 0.81, due to the two-step etching process that increases modal confinement to the gain layer by undercutting the InP pedestals. The laser is surrounded by a dielectric shield of Al2O3 of thickness tshield =168 nm, which is in turn surrounded by a metal-cladding layer (silver) around the gain region. In this laser design, the distance rcontact1 of the top contact wire from the center of the laser is 20 μm, while the bottom contact wire is a far enough distance rcontact2 >> rcontact1 from the laser’s center that it does not play a role in heat dissipation. In Sections C.2−C.6, we calculate the various self-sources, which are briefly introduced in Section 8.1.

284

Appendix C: Modeling Thermal Effects in Nanolasers

Table C.1 Dimensions and material parameters of the nanolaser analyzed. Conductivities are calculated from the doping level and carrier mobility using Equation (C.3).

Layer

Material

Doping (cm−3)

Top contact Upper pedestal top Upper pedestal bottom Gain Lower pedestal top Lower pedestal bottom Bottom contact

InGaAs, nInP, n InP, n InGaAs InP, p InP, p InGaAsP, p+

2e19 5e18 1e18 − 1e18 5e18 2e19

Thickness Radius (nm) (nm)

Carrier mobility (cm2∙V−1∙s−1)

Electric conductivity (S/m)

125 235 235 300 125 725 135

2.5e3 1.25e3 2e3 − 80 35 50

8.001e5 1.001e5 3.204e4 − 1.282e3 2.803e3 1.602e4

574 358 358 574 431 431 N/A

Figure C.1

Diagram of electrically pumped nanolaser to be analyzed. The laser has an InGaAs gain region of radius rcore surrounded by a lower InP plug of radius rlower and upper InP plug of radius rupper. The laser is surrounded by a dielectric shield of amorphous Al2O3 (α-Al2O3), of thickness tshield, which is in turn surrounded by a metal-cladding layer (silver) of thickness tcladding. The metal-cladding layer connects the laser’s InGaAsP top contact layer to the top electrical contact wire, at a distance rcontact1 from the laser’s center. The bottom contact InGaAsP layer is connected to the bottom electrical contact wire at a distance rcontact2 >> rcontact1.

C.2

Ohmic (Joule) Heating Using a Simple Stack Model Joule heating is self-heating due to the resistance of each of the semiconductor layers, and is given by QJ ¼ I 2 Rs

ðC:1Þ

where I is the operating current and Rs is the stack resistance of the semiconductor layer. The stack resistance of the ith layer may be calculated from the

Appendix C: Modeling Thermal Effects in Nanolasers

285

layer’s radius ri, thickness ti, and conductivity σi using the standard formula for stack resistance: Ri ¼

ti

ðC:2Þ

σ i πðri Þ2

The material conductivity of the ith layer may be calculated using σ i ¼ ni μi qe

ðC:3Þ

where ni is the doping level, μi is the carrier mobility, and qe is the electron charge. The bottom contact layer behaves like cylindrical thin film contact geometry. The resistance in this layer is given by



 rlp σ bc 1 rbc 1 Rbc ¼ ln Rc ; þ ðC:4Þ 2πσ bc tbc 4σ bc rlp rlp tbc σ lp where the first term is the resistance of the bottom contact layer region between the nearest contact wire and the laser’s lower pedestal, and the second term is the resistance of the bottom contact layer directly underneath the laser’s lower pedestal; rlp is the radius of the lower pedestal, and rbc is the distance between the laser’s center and the nearest contact wire. In the laser example, rlp = rlower and rbc = rcontact1 as that schematically shown in Figure C.1; σbc is the conductivity of the bottom contact layer, while σlp is the conductivity of the laser’s lower pedestal. An empirical expression for R c is numerically found to be

Rc

rlp σ bc ; tbc σ lp



 r

 Δ lp rlp tbc  ffi R c0 þ 2 tbc

2σ bc

 rlp σ bc þ β σ lp tbc

ðC:5Þ



 rlp rlp rlp where R c0 and Δ are defined differently depending on the ratio . For tbc tbc tbc rlp ≤ 1 (lower pedestal radius is less than bottom contact thickness) 0:0011 ≤ tbc

3





2 rlp rlp rlp rlp R c0  61773 ¼ 1  22968 þ 49412 tbc tbc tbc tbc

4

5 rlp rlp þ 3811  08836 ; tbc tbc





2 rlp rlp þ 00073 ¼ 00184 þ 0:0808 ðC:6Þ tbc tbc

 rlp While for 1 < < 10 (lower pedestal radius is larger than bottom contact tbc thickness), Δ

rlp tbc



286

Appendix C: Modeling Thermal Effects in Nanolasers

R c0

rlp tbc





rlp ¼ 0295 þ 0037 tbc

1

2 rlp þ 0:0595 ; tbc

 rlp ¼ 00409x4  01015x3 þ 0265x2  00405x þ 0:1065 tbc

 rlp where x ¼ ln : tbc

ðC:7Þ

Δ

For both cases β



2

 rlp rlp rlp þ 00949 ¼ 00016 þ 0:6983 tbc tbc tbc

ðC:8Þ

rlp 431 ¼ 3:2, yielding ¼ tbc 135 R c0 ¼ 0:312, Δ = 0.333, β = 0.715, leading to R c ¼ 0:64 using the previous expressions. Therefore, the second term in Equation (C.5), the contribution to the bottom contact resistance by the region just below the laser pedestal, is 23 Ω. This is small compared to the first term of Equation (C.5), the contribution by the rest of the bottom contact, which is 282 Ω. Since in most nanolaser geometries rbc ≫rlp , the first term of Equation (C.5) will be much larger than the second term. In the thermal simulation, the total resistance from both terms of Equation (C.5), 305 Ω, is distributed across the entire bottom contact layer. For greater accuracy, the resistive heating resulting from the second term can be modeled as located directly beneath the laser pedestal, while the resistive heating resulting from the first term can be distributed across the rest of the bottom contact. The Joule heat sources for the example nanolaser are listed at the left of Figure C.3, and contribute a total of 0.226 mW of heating power to the nanolaser. The largest source of Joule heating is the lower pedestal, followed by the bottom contact. The bottom contact is expected to contribute negligibly to laser heating, since the intensity of heat generated is low, and the heat can easily flow out of this region into the substrate and bottom contact wire. Similarly, the lower pedestal is adjacent to the bottom contact, which can easily remove heat from the laser pedestal. For the laser whose geometry is listed in Table C.1,

C.3

Junction Heating Junction heating is the heat generated by the voltage change at the junction between the undoped gain layer and the adjacent doped semiconductor layers. To calculate the voltage changes, the device’s electrical behavior should be simulated; SILVACO’s ATLAS simulation yields voltage, carrier density, and quasi-Fermi level separation. Using the voltage change Vjn at the nth junction, the power dissipated will be Pjn = Ith∙Vjn, where Ith is the laser threshold operating current.

Appendix C: Modeling Thermal Effects in Nanolasers

287

Figure C.2

Result of electronic simulation. (a) Potential difference as a function of vertical distance from the top of the top contact. Length = 0 corresponds to the top of the top contact layer, and length = 1.88 μm corresponds to the bottom of the bottom contact layer. (b) Carrier density as a function of injection current. (c) Quasi-Fermi-level (QFL) separation as a function of injection current.

Figure C.3

Amount and location of heating sources in the example nanolaser, at 300 K ambient temperature and 0.5 mA operating current. Each region is according to its thermal conductivity at 300 K.

For the laser described in Figure C.1 and Table C.1, the device voltage, carrier density, and I-V curve are shown in Figure C.2(a), (b), and (c), respectively. The potential difference at each junction between differently doped layers is visible in Figure C.2(a). In this example, a threshold current of Ith = 0.4 mA and a slightly larger operating current of I = 0.5 mA are used, which were experimentally used for a fabricated nanolaser with dimensions similar to this example. Figure C.3 lists the calculated junction heating sources for this laser, located at junctions 3 and 4 (all other junctions in the laser contribute to heterojunction heating instead, as described in Section C.4). Together these two junctions contribute 0.110 mW of self-heating.

288

Appendix C: Modeling Thermal Effects in Nanolasers

C.4

Heterojunction Heating Similar to junction heating, heterojunction heating is the heat generated by the voltage change at the remaining junctions, between the doped semiconductor layers. As before, the power dissipated at the nth junction is Pjn = I∙Vjn. This time, the current used is the operating current, rather than threshold current. Figure C.3 lists the calculated heterojunction heating sources for the laser described in Figure C.1 and Table C.1. Heterojunction heating adds 0.678 mW of self-heating to the laser. Most of this heating, however, takes place at the junction between the pedestals and the top or bottom contacts and is easily dissipated via the contacts.

C.5

Surface recombination heating Surface recombination is an additional heating term that is not usually considered for larger lasers but becomes important for small lasers, for which the ratio of surface area to volume is large. The rate of surface recombination Us in the gain region is given by Us ¼

n τs

ðC:9Þ

where n is carrier density and τs is carrier lifetime. The carrier lifetime is given by 1 Aactive ¼ υs τs Vactive

ðC:10Þ

where Aactive and Vactive are the area and volume of the gain region, and νs is the surface recombination velocity. To calculate νs at 300 K for InGaAs, we use the value of νs at 77 K, νs = 6.7×103 cm/s, along with the knowledge that the νs is proportional to the square root of temperature. Thus, at 300 K, pffiffiffiffiffiffiffiffi 300 υs ð300KÞ ¼ υs ð77KÞ pffiffiffiffiffi ¼ 1:3  104 cm=s 77

ðC:11Þ

Using the previous calculations to get the surface recombination rate Us, we can then use the simulation of the QFL from Figure C.2(c) to calculate the heating power generated from surface recombination, Ps ¼ Us  Vactive  QFL

ðC:12Þ

For the example nanolaser operating at T = 300 K and at injection current I = 0.5 mA, QFL separation is 1.14 eV and carrier concentration is 7.07×1018 cm−3. Using Equation (C.12), the surface recombination heating is calculated to be 0.393 mW.

Appendix C: Modeling Thermal Effects in Nanolasers

C.6

289

Auger Recombination Heating The last heating source we consider is Auger recombination, which, like surface recombination heating, becomes a source of heat in the gain region. The Auger recombination rate UA is given by UA ¼ An3 Vactive

ðC:13Þ

where n is the carrier density, Vactive is the volume of the gain region, and A is the Auger coefficient. For InGaAs at 300 K, the Auger coefficient is 9.8×10−29 cm6/s. Using the carrier density for the example laser at an injection current of 0.5 mA as calculated in Figure C.2(b), Auger recombination rate UA is calculated to be 1.075×1016 s−1. The Auger heating is then calculated as PA ¼ UA  QFL

ðC:14Þ

which yields 1.963 mW. This is by far the largest source of self-heating for the nanolaser; because this heat source is located in the middle of the semiconductor stack, it will also be the most difficult to dissipate. In Figure C.3, the various self-heating sources for the example laser are summarized and their locations in the semiconductor stack shown. The junction and heterojunction heating sources are implemented as area heating sources located at the interfaces between layers, while Joule, surface recombination, and Auger recombination heating are volume heating sources implemented as distributed within each semiconductor layer. Only half of the device cross section is shown since the device is approximated to be axially symmetric. Note that each of these heat sources is dependent on operating temperature and on injection current. For the most accurate reflection of nanolaser temperature behavior, these heat sources should be updated to reflect the changing temperature as the nanolaser self-heats. Compared to the other heating sources shown in Figure C.3, Joule heating is a minor contribution to laser self-heating.

Appendix D Constriction Resistance and Current Crowding in Nanolasers

D.1

Vertical Contact Structure For electrically pumped nanolasers, because of the limited contact area between an electrode and the cavity, implementing low-resistance and Ohmic electrodes is of crucial importance in minimizing self-heating. This is in part done in nanofabrication by choosing the optimal thicknesses of contact metals and performing rapid thermal annealing after metal deposition. Additionally, adequate understanding of the performance of electrodes in a nanolaser can be of great aid in understanding the laser performance and, in so doing, aid in the optimization of cavity design. In electrical simulations, Ohmic contact and uniform current spreading into the semiconductor material are usually assumed, as we saw in the examples in Chapter 8. In reality, however, not only is the contact resistance not negligible, neither is the current spreading uniform. In particular, the geometrical shape of the semiconductor material will affect both these parameters, and it is expected that some trade-off exists between electrical and optical mode performance. In this appendix, without loss of generality, we examine nanopillar cavities without and with InP undercut, similar to those seen in Sections 7.2 and 7.3 of Chapter 7. These nanolasers are schematically shown in Figure D.1(a) and (b), respectively. In the case of an undercut design, Figure D.1(b) presents the ideal vertical InP pedestals sidewall (noted as 0 degree). In practice, sidewall angles can vary from −20 to 20 degrees due to fabrication variations. Section 8.3 of Chapter 8 analyzed how the various pedestal sidewall angles affect the nanolaser’s optical and thermal properties assuming the ideally Ohmic electrodes and uniform current spreading (Section 8.1), but it is not clear how the sidewall angle would affect the contact resistance and current spreading, which will in turn affect the optical and thermal performance of the laser. The latter issue motivated a more in-depth study – the focus of this appendix − to provide quantitative analysis to the thermal management of electrically pumped nanolasers, where self-heating is believed to be one of the main failure mechanisms. Typically, a nanolaser has a small-area top electrode that is in direct contact with the optical cavity and is comparable in size to the laser dimension (schematically shown in Figure D.1), and current flows vertically from the top electrode into the laser. It also has a large-area bottom electrode that can be shared among a number of devices on the same

Appendix D: Constriction Resistance and Current Crowding in Nanolasers

291

Figure D.1

Schematic of a nanolaser (a) without and (b) with InP undercut, with both electrodes being in direct contact with the cavity. Reprinted from reference [356] with permission from Institute of Electrical and Electronics Engineers (IEEE).

Figure D.2

Schematic of a realistic nanolaser with only one electrode in direct contact with the cavity. Reprinted from reference [356] with permission from Institute of Electrical and Electronics Engineers (IEEE).

chip (schematically shown in Figure D.2), and current flow into the bottom of the laser can be either horizontal or vertical, depending on the resistances of the various substrate layers. Sometimes, both electrodes can be small-area electrodes in direct contact with the optical cavity (Figure D.1) or large-area electrodes that are remotely connected to the optical cavity, under which condition the same contact resistance and current spreading analysis can be applied to both electrodes. To study the vertical contact structure that is typical for the top electrode in nanolasers, which is above the gain region in nanolasers of Figure D.3(a), let’s consider the simplified geometry of the vertical contact, shown in Figure D.3(b). Regions I and II may represent the upper pedestal region and the top contact region in Figure D.3(a), respectively. Similar to the study of optical performance of realistic undercut nanolasers in Section 7.1 (Figure 7.9), the angles of the sidewall tilt (counterclockwise to the vertical axis) of regions I and II are denoted by θ1 and θ2, respectively. Region I has average radius r1, thickness h1, and resistivity ρ1, whereas region II has average radius r2, thickness h2, and resistivity ρ2. The average radius of region II (top electrode) r2 is usually assumed to be the same as that of the gain region (Figure D.3(a)). The total resistance from AB to EF in Figure D.3(b) is R ¼ RI þ Rc þ RII

ðD:1Þ

292

Appendix D: Constriction Resistance and Current Crowding in Nanolasers

Figure D.3

(a) Nanolaser to be analyzed in 2D sideview. (b) Vertical contact above the gain region (dashed box region in (a)) with tilted sidewalls, with both angles of sidewall tilt θ1 and θ2 (both positive as shown). The z-axis is the axis of rotation for the cylindrical geometry. Reprinted from reference [356] with permission from Institute of Electrical and Electronics Engineers (IEEE).

where RI and RII are the bulk resistance of region I from DG to EF and region II from AB to CH, respectively. They are defined as RI ¼

ρ 1 h1 1 ; 2 2 π r1  ðh1 =4Þtan2 θ1

ρ 2 h2 1 ; π r22  ðh22 =4Þtan2 θ2 ρ Rc ¼ 2 R c 4a

RII ¼

ðD:2Þ

where Rc is the additional constriction resistance due to current crowding effects at the interface between regions I and II in Figure D.3(b), where a = r1–0.5h1 tan θ1 is the radius of contact interface OG. Note that if θ1 and θ2 become zero, there would be no sidewall tilt in the contact members and Equation (D.2) would reduce to that of cylinders. For the case of zero sidewall tilt, the normalized constriction resistance Rc can be calculated exactly as in reference [357]. It is

Rc

r1 h1 r1 ρ1 ; ; ; r2 r1 h2 ρ2

 ¼

∞ 8 ρ1 X J1 ð β n r 1 Þ Bn sinhðβn h2 Þ π ρ2 n¼1 βn r 1

ðD:3Þ

for arbitrary values of r1, r2 (r2 > r1), h1, h2 and ρ1/ρ2, where Bn can be calculated following reference [357] (Equation (B4) within), βn satisfies J1(βnr2) = 0, and J1(x) is the Bessel function of order one. In the more general case, when θ1 and θ2 are non-zero, instead of using an analytical expression, we obtain the constriction resistance Rc from Equations (D.1) and (D.2) after knowing the total resistance R by the voltage-to-current ratio, for the structure in Figure D.3(b). The parameters used are those for the

Appendix D: Constriction Resistance and Current Crowding in Nanolasers

Figure D.4

293

Constriction resistance Rc for vertical contact structure in Figure D.3, as a function of (a) amount of undercut = (r2 – r1)/r2, for the case of zero sidewall tilt θ1 = θ2 = 0, for r2 = 775 nm, 550 nm, and 225 nm; (b) angle of sidewall tilt of region I θ1, for r2 = 550 nm, with amount of undercut = 0%, 20%, and 50%; (c) θ1, with amount of undercut = 20%, for r2 = 775 nm, 550 nm, and 225 nm; (d) angle of sidewall tilt of region II θ2, for the case of r2 = 550 nm with amount of undercut = 20%. Symbols are for numerical data, while dashed lines in (a) for analytical calculations from Equations (D.2) and (D.3), and solid lines for curve connecting the numerical data. The dash-dotted lines are for the ratio of constriction resistance to total resistance Rc /Rtot. In (d), Rc/Rtot ≈ 0.6%. Reprinted from reference [356] with permission from Institute of Electrical and Electronics Engineers (IEEE).

electrically pumped nanolasers studied in Chapter 7: h1= 470 nm, h2 =125 nm, ρ1 = 1/σ1 = 1/1.001 × 105 S/m = 1 × 10-5Ω m, ρ2 = 1/σ2 = 1/8.011 × 105. S/m = 1.248 × 10−6 Ω m (see Table C.1). Figure D.4(a) shows the constriction resistance Rc as a function of the amount of undercut, defined as (r2 – r1)/r2, for different radius r2 = 775 nm, 550 nm, and 225 nm, for the case of zero sidewall tilt. It is clear that Rc increases as the amount of undercut increases, indicating more severe current crowding at the interface. The effect of undercut on the constriction resistance is more profound for a smaller gain radius. The numerical calculations are compared with our analytical results obtained from Equations (D.2) and (D.3) for θ1 = θ2 = 0. As seen from Figure D.4(a), excellent agreement is obtained. Figure D.4(b) shows Rc as a function of the angle of sidewall tilt θ1 of region I for the case r2 = 550 nm, under different amounts of undercut. Figure D.4(a) shows Rc as a

294

Appendix D: Constriction Resistance and Current Crowding in Nanolasers

function of θ1 for an fixed undercut = 20%, for different radius r2. The constriction resistance Rc increases significantly as θ1 deviates from zero. For larger amounts of undercut (Figure D.4 (b)) or smaller radius (Figure D.4(c)), the increase of Rc is even more sensitive to the region I sidewall tilt angle θ1. The sidewall tilt of region II θ2 is also expected to increase the constriction resistance Rc. However, since region II is highly conductive compared to region I, ρ2/ρ1 ~ 0.1, the increase of Rc with θ2 is almost negligible, as shown in Figure D.4(d). It is important to note that the constriction resistance is typically smaller than the bulk resistance of regions I and II, Rc < RI,RII. For the cases studied in Figure D.4, the ratio of constriction resistance to total resistance Rc/Rtot is 16% at most. However, the current density and heating near the current constricted region are significantly higher than those in the bulk region of I and II, as discussed later. While the value of constriction resistance Rc contributes to total resistance due to current crowding, it is important to identify where the most current crowding occurs in a contact structure. Figure D.5 shows the current density distribution J (r,z) for the vertical contact structure in Figure D.3(b) with the sidewall angle tilt θ1 = 30°, 0°, and −30°, for r2 = 550 nm and amount of undercut = 50%. In the calculation, we assume a total injection current of 0.5mA, corresponding to a current density of J0 = 2.1 nA/nm2 in region I with zero sidewall tilt if uniform current distribution is assumed. As shown in Figure D.5(a), when θ1 = 30°, severe current crowding occurs at the constriction corner point G, with a current density as high as 23.5 nA/nm2 at 1 nm away from point G toward the center axis (right at point G the current density is infinity due to the mathematically sharp corner assumed), which is about 11 times J0. On the other hand, the current density near the bottom rim of region I (point F) is suppressed. When θ1 = 0° (Figure D.5(b)), the current crowding region becomes significantly smaller compared to that of θ1 = 30°, with a current density of 4.5 nA/nm2 (~2J0) at 1 nm inward from point G along the interface. When θ1 = −30° (Figure D.5(c)), the current density is mostly crowded near the bottom surface of region I, with current density of 18.9 nA/nm2 (~ 9J0) at 1 nm inward from point F along the interface, whereas the current density near the interface between regions I and II becomes quite uniformly distributed. It is important to note that even though the constriction resistance Rc = 3.39Ω for θ1 = −30° and Rc = 4.46Ω for θ1 = 30° is comparable (cf. Figure D.5(b)), the location where current crowding occurs is very different. Figure D.5(d)−(f) shows the corresponding distribution of Joule heating power per volume Pðr; zÞ ¼ ρðr; zÞJ 2 ðr; zÞ. The total Joule heating power in the structures in Figure D.5(d)−(f) is 7.7, 5.1, and 7.4 μW, respectively, corresponding to an increase of 51% for θ1 = 30° and 45% for θ1 = −30° relative to the straight sidewall case θ1 = 0°. Though the total power is not increased much by sidewall tilt, the power density near the most current-crowded region is significantly increased. The hottest spot follows the region of highest current density closely, yielding P ≈ 3.4, 0.085, and 3.8 pW/nm3 at 1nm away from the hottest corner for θ1 = 30°, 0° and −30°, respectively, corresponding to an increase of P by 39 times for θ1 = 30° and by 44 times for θ1 = −30° relative to the case θ1 = 0°. Figure D.6 shows the magnitude of current density along the contact interface OH for cases in Figure D.5(a)−(c) with different angle of tilt.

Appendix D: Constriction Resistance and Current Crowding in Nanolasers

295

Figure D.5 (a)–(c) Current density distribution J ðr; zÞ, and (d)–(f) the corresponding Joule heating power per

volume Pðr; zÞ ¼ ρðr; zÞJ 2 ðr; zÞ, for the vertical contact structure in Figure D.3 for r2 = 550 nm and amount of undercut = 50%. (a) and (d) are for the sidewall angle tilt θ1 = 30, (b) and (e) for θ1 = 0, and (c) and (f) for θ1 = −30. In the calculation, we assume a total injection current of 0.5 mA. Reprinted from reference [356] with permission from Institute of Electrical and Electronics Engineers (IEEE).

D.2

Horizontal Contact Structure In similar ways, the constriction resistance and current crowding can be calculated for horizontal contacts, such as the bottom contact of the electrically pumped nanolaser of Chapter 7, as depicted in Figure D.7. Regions I and II of Figure D.7(b) represent the nanolaser’s lower pedestal region and the bottom thin film region (Figure D.7(a)), respectively. Similar to Appendix D.1, we define the angle of the sidewall tilt in region I to be θ1. Region I has average radius r1, thickness h1, and resistivity ρ1, whereas region II has average radius r2 (r2 ≫ r1), thickness h2, and resistivity ρ2.

296

Appendix D: Constriction Resistance and Current Crowding in Nanolasers

Figure D.6

The magnitude of current density along OH for cases in Figure D.5 (a)–(c). Reprinted from reference [356] with permission from Institute of Electrical and Electronics Engineers (IEEE).

Figure D.7

(a) Nanolaser to be analyzed. (b) Horizontal contact below the gain region (dashed box region in (a)) with tilted sidewall of angle θ1 (positive as shown). The radius of the gain region above region I is rg. For zero undercut, r1 = rg. The z-axis is the axis of rotation for the cylindrical geometry. Reprinted from reference [356] with permission from Institute of Electrical and Electronics Engineers (IEEE).

The total resistance from EF to BC (and AH) in Figure D.7(b) is R ¼ RI þ Rc þ RII

ðD:4Þ

where RI is the bulk resistance of region I from EF to DG, and RII is the bulk resistance of the bottom thin film disk in region II from DK to CB (and from GJ to HA), which are respectively defined as

Appendix D: Constriction Resistance and Current Crowding in Nanolasers

ρ1 h1 1 ; 2 2 π r1  ðh1 =4Þtan2 θ1

 ρ2 r2 RII ¼ ln ; 2πh2 r1 þ ðh1 =2Þtanθ1 ρ Rc ¼ 2 R c 4a

297

RI ¼

ðD:5Þ

which includes the remaining resistance in the region GDJK, and more importantly the constriction resistance [358] due to current-crowding effects at the interface between regions I and II in Figure D.7(b), where a = r1 + 0.5h1 tan θ1 is the radius of contact interface OD. Note that if θ1 becomes zero, there would be no sidewall tilt in region I and Equation (D.4) would reduce to that of cylinders. For the case of zero sidewall tilt, the normalized constriction resistance Rc is calculated exactly as in reference [357] and is given by





 ∞ r1 h1 r1 ρ1 8 ρ1 X λn h2 J1 ðλn r1 =r2 Þ 2r1 r2 Rc ; ; ; Bn coth  ln ¼ π ρ2 n¼1 λn r1 =r2 r2 r1 h2 ρ2 r2 πh2 r1

ðD:6Þ

for arbitrary values of r1, r2 (> r1), h1, h2, and ρ1/ρ2, where Bn is calculated from equation (C4) of reference [357], λn satisfies J0(λn) = 0, and J0(x) and J1(x) are the Bessel function of order zero and one, respectively. In numerical calculations, the total resistance R for the structure in Figure D.7(b) is obtained by the voltage-to-current ratio. We can then obtain the constriction resistance Rc from Equations (D.4) and (D.5). Using dimensions and resistivities in typical nanolasers based on epitaxially grown wafer stack [359], we fix the following parameters in Figure D.7(b) as h1 = 600 nm, h2 = 135 nm, ρ1 = 1/σ1 = 1/2.803 × 103 S/m = 3.568 × 10−4 Ω m, ρ2 = 1/σ2 = 1/1.602 × 104 S/m = 6.2422 × 10−5 Ω m. The bottom electrode is located at a distance far away from the nanolaser (Figure D.2), typically on the order of tens of μm. In this calculation, we fix r2 = 2μm in Figure D.7(b), because the current flow lines would become almost uniformly distributed at a distance beyond 2r1 from the contact constriction corner, when the contact dimension is larger than the bottom thin film thickness, OD ≥ h2 [357]. Without any undercut, the average radius of region I should be the same as that of the gain region above region I (not shown in Figure D.7(b)), that is, r1 = rg for zero undercut (Figure D.1(a)). With undercut, r1 is smaller than rg, the amount of undercut here is defined as (rg − r1)/rg. Figure D.8(a) shows the constriction resistance Rc as a function of the amount of undercut, for different gain radius rg = 775 nm, 550 nm, and 225 nm, for the case of zero sidewall tilt. Rc increases as the amount of undercut increases, indicating more severe current crowding at the interface. Similar to that seen for the vertical contact (Figure D.4(a)), the effect of undercut on the constriction resistance is more profound for smaller gain radius. In reported electrically pumped nanolasers [37, 38], the semiconductor materials above the gain region are always n-doped and those below the gain region p-doped. As a result of the

298

Appendix D: Constriction Resistance and Current Crowding in Nanolasers

Figure D.8

Constriction resistance Rc for horizontal contact structure in Figure D.7b, as a function of (a) amount of undercut = (rg – r1)/rg, for the case of zero sidewall tilt θ1 = 0°, for rg = 775 nm, 550 nm, and 225 nm; (b) angle of sidewall tilt of region I θ1, for rg = 550 nm, with amount of undercut = 0%, 20%, and 50%; (c) θ1, with amount of undercut = 20%, for rg = 775 nm, 550 nm, and 225 nm. Symbols are for the numerical data, dashed lines in (a) for analytical calculations from Equations (D.5) and (D.6), and solid lines for curve connecting the numerical data. The dash-dotted lines are for the ratio of constriction resistance to total resistance Rc/Rtot. Reprinted from reference [356] with permission from Institute of Electrical and Electronics Engineers (IEEE).

larger resistivity of p-doped materials in the bottom pedestal and bottom contact thin film, the constriction resistance is much larger compared to that in the top vertical contact (Figure D.7(a) vs. Figure D.4(a)). This is consistent with the previous study, which reveals that the largest source of Joule heating is the lower pedestal, followed by the bottom contact [359]. The numerical calculations are then compared with analytical results obtained from Equations (D.2) and (D.3). Excellent agreement is obtained, as shown in Figure D.7(a). Figure D.7(b) shows Rc as a function of angle of sidewall tilt θ1 of region I for the case rg = 550 nm, under different amount of undercut. Figure D.7(c) shows Rc as a function of θ1 for an undercut = 20%, for different gain radius rg. The constriction resistance Rc increases significantly as θ1 deviates from zero. Similar to the vertical contact, for larger amounts of undercut (Figure D.7(b)) or smaller radii (Figure D.7(c)), the increase in Rc is even more sensitive to θ1. The constriction resistance is also typically smaller than the bulk resistance

Appendix D: Constriction Resistance and Current Crowding in Nanolasers

299

of regions I and II, Rc < RI,RII. For the cases studied in Figure D.7, the ratio of constriction resistance to total resistance Rc/Rtot is 16% at most. However, the current density and heating near the current constricted region are significantly higher than those in the bulk region of I and II. It is important to examine where the most current crowding occurs. Figure D.8(a)−(c) shows the current density distribution J ðr; zÞ for the horizontal contact structure in Figure D.6(b) of the sidewall angle tilt θ1 = 30°, 0°, and −30°, respectively, for rg = 550 nm and amount of undercut = 50%. With zero sidewall tilt and uniform current distribution, the assumed injection current of 0.5 mA corresponds to a current density of J0 = 2.1 nA/nm2 in region I. As shown in Figure D.8(a), when θ1 = 30°, the current density is mostly crowed near the top boundary of region I, with current density as high as 28.5 nA/nm2 at 1 nm inward from the rim point E, which is about 14 times J0, and further; it is larger compared to that of vertical contact (Figure D.5(a)). On the other hand, the current density near the interface between regions I and II becomes quite uniformly distributed. When θ1 = 0° (Figure D.9(b)), the current density distribution near the top boundary of region I becomes almost uniform. The most current-crowded region is near the constriction corner, point D, with a current density of 5.69 nA/nm2 (~ 3J0) at 1 nm inward from point D along the interface. When θ1 = −30° (Figure D.9(c)), the current-crowding region near the constriction corner D is greatly increased, with a current density as high as 28.4 nA/nm2 at 1 nm

Figure D.9

(a)–(c) Current density distribution J ðr; zÞ, and (d)–(f) the corresponding Joule heating power per volume Pðr; zÞ ¼ ρðr; zÞJ 2 ðr; zÞ, for the vertical contact structure in Figure D. 7 for r2 = 550 nm and amount of undercut = 50%. (a) and (d) are for the sidewall angle tilt θ1 = 30°, (b) and (e) for θ1 = 0°, and (c) and (f) for θ1 = −30°. In the calculation, we assume a total injection current of 0.5 mA. Reprinted from reference [356] with permission from Institute of Electrical and Electronics Engineers (IEEE).

300

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Figure D.10

The magnitude of current density along ODC for cases in Figure D.5(a)–(c). Reprinted from reference [356] with permission from Institute of Electrical and Electronics Engineers (IEEE).

inward from point D, which is about 14 times J0, much larger compared to that of vertical contact (Figure D.5(c)). The current density near point E is suppressed. The magnitude of current density along the contact interface ODC is shown in Figure D.5(g) for cases with different angle of tilt. It is important to note that even though the constriction resistance Rc = 229Ω for θ1 = 30° and Rc = 268Ω for θ1 = −30° are comparable (cf. Figure D.4(b)), the location where current crowding occurs is very different. Figure D.5(d)−(f) shows the corresponding distribution of Joule heating power per volume Pðr; zÞ ¼ ρðr; zÞJ 2 ðr; zÞ. The total Joule heating power in the structures in Figure D.5 (d)−(f) is 458.8, 268.6, and 495.3 μW, respectively, corresponding to an increase of 71% for θ1 = 30° and 84% for θ1 = −30° relative to the straight sidewall case θ1 = 0°. The power density near the most current-crowded region is significantly increased by the sidewall tilt. The hottest spot follows closely the region of highest current density, yielding P = 250, 4.74, and 421 pW/nm3 at 1 nm away from the hottest corner for θ1 = 30°, 0°, and −30°, respectively, corresponding to an increase of P by 52 times for θ1 = 30° and by 88 times for θ1 = −30° relative to the case θ1 = 0°. Figure D.10 shows the magnitude of current density along the contact interface ODC for cases in Figure D.5(a)−(c) with different angle of tilt. Note that the power density near the hottest spot is about one order of magnitude higher compared to that of vertical contact in Appendix D.1, for the same amount of undercut and same angle of sidewall tilt. While designs with θ1 of 30° and −30° produce power densities on the same order of magnitude, a design with θ1 = −30° would affect the overall heating less for the nanolaser structure shown in Figure D.2, because the local heat can be quickly dissipated through the large-area substrate rather than through the gain region.

D.3

Discussion In this appendix, we studied the constriction resistance and current crowding in nanolasers by examining both vertical contact and horizontal contact structures. The former is

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301

formed above the gain region, between the upper contact region and the upper pedestal region; whereas the latter is formed below the gain region, between the lower pedestal region and the bottom thin film layer. For the nanolaser structure under investigation, constriction resistance and the degree of current crowding in the bottom horizontal contact in nanolasers are much larger than those in the top vertical contact. The effects of nanolaser radius, amount of undercut, and angle of sidewall tilt are studied in detail. For both contact scenarios, constriction resistance increases when the nanolaser radius decreases, the amount of undercut increases, or the angle (either positive or negative) of sidewall tilt increases. The current density distribution and the corresponding Joule heating power density distribution are calculated, and the location where most current crowding and most Joule heating occur is identified. In the top vertical contact, when the pedestal sidewall tilt angle is positive, the hottest spot is located near the constriction corner at the interface between the top contact and the upper pedestal (points D and G in Figure D.3(b)); when the tilt angle is negative, the hottest spot is located near the bottom rim of the upper pedestal (points E and F in Figure D.3(b)). Because Joule heating in proximity to the gain region degrades the performance of gain medium, it is therefore preferable to have a positive tilt angle for the top contact. In the bottom horizontal contact, when the sidewall tilt is positive, the hottest spot is near the top rim of the lower pedestal region (points E and F in Figure D.7(b)); when the sidewall tilt is negative, the hottest spot is near the constriction corner between the lower pedestal and bottom thin film (points D and G in Figure D.7(b)). In the example nanolaser design in which only one pedestal sidewall tilt angle can be chosen, one needs to evaluate the opposite preferences for the upper and lower contacts. First, at any given tilt angle, the upper vertical contact produces less heat power density than that from the lower horizontal contact; second, at any given power density, the larger heating from the lower contact can be efficiently removed via the substrate if it is generated close to the substrate. Negative tilt angle is thus preferred in order to reduce Joule heating in proximity to the gain region. Future work may investigate the effects of contact resistance and current crowding on the threshold gain and efficiency of nanolasers. Because the bottom electrode is typically attached only to one side of the nanolaser (Figure D.2), it is important to study the asymmetric injection of charge carriers in nanolasers, which is expected to increase the resistance of the bottom contact and thin film significantly [360]. Coupled thermalelectrical conduction with temperature-dependent material properties is also of future interest.

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Index

3 dB bandwidth, 247, 252 active material, 7 active medium, 7, 132. See gain medium bulk, 7 multiple quantum well, 7 quantum dot, 7 surface-to-volume ratio, 171 volume, 134 adhesion layer, 160 aluminum nitride, 178 aluminum oxide, 154 amplified spontaneous emission, 62, 212 atomic layer deposition (ALD), 174, 266 axial symmetric, 136 Bardeen-Cooper-Schriefer (BCS), 221 Bessel function, 39, 292, 297 bit-error rate, 253 Bloch plasmon-polariton (BPP), 93 Bohr radius, 221 Bose-Einstein condensate, 214, 216 boundary-value problem, 43 Bragg period, 2 Bragg grating, 256, 257 bulk gain, 133, 135, 145, 249, 250 cavity linewidth, 73, 248 mode, 13, 80 transparent condition, 66, 73, 75 Cavity-free nanolaser, 26, 202 cavity-QED, 66, 312 CCD camera, 59, 61, 62, 160 chip-scale integration, vii, 3, 9, 30, 37, 54, 119, 123, 132, 168, 231, 245, 256, 263, 266, 317 clamp carrier, 144 current, 171 clamping carrier, 142

cloud computing, 231 CMOS compatible, 54, 118, 233, 234 coherence, 10, 63, 89, 261 first order. See spatial coherence phase, 212 spatial, 59, 62, 63, 227 colloidal QD, 9 composite gain waveguide, 38 confinement factor, 6, 11, 16, 18, 21, 72 energy confinement factor, 13 power confinement factor, 13 power confinement factor, electric field confinement factor, electric energy confinement factor, 16 constriction resistance, 200, 292 core-shell, 23, 26, 99 coupling coefficient, 257 coupling efficiency, 131, 239, 263 current crowding, 200, 292 cutoff frequency, 127 data transmission, 93, 231, 232 data-rate ratio, 254 density of states electronic, 6, 136, 138, 223, 279 free space, 114 photonic, 6, 212 reduced, 67 dielectric constant, 120, 203 diffraction limit, 10, 29, 93 diffusion, 55, 58, 138, 139 length, 71 time, 71 direct-transition model, 67 dispersion relation, 27, 39, 203, 265 k-point, 27, 202 distributed Bragg reflector, 20, 219 double heterostructure, 99, 133, 148, 159, 162 drift, 138, 261 Drude model, 81, 92, 191 dry etch, 58, 160, 165, 200 reactive ion etching, 51

322

Index

e-beam evaporation, 160 e-beam lithography, 58, 123, 159, 178 edge-emitting, 235, 261 effective index, 10, 92, 98, 157 mismatch, 157 effective mass, 218, 278 reduced, 279 effective medium theory, 98, 265 effective mode volume, 3, 16 electrical insulation, 37, 147, 162, 177, 178, 246 electro-luminescence, 31, 162 emitter-field-reservoir model, 66 energy confinement factor. See confinement factor epitaxial, 25, 177, 227, 235, 259 epitaxially grown, 9, 57 evanescent, 127, 235, 242 evanescent-coupling, 242 exciton-polariton laser, 10, 215 inversionless laser, 133 excitons, 133, 215, 216, 221 Fabry-Perot cavity, 10, 12, 21, 22, 29, 146, 165 Fermi distribution, 75, 253 field antinode, 71, 227 field node, 71, 274 finite element method (FEM), 41 finite-difference time-domain (FDTD), 98 footprint, 3, 29, 30, 149, 192, 235, 263 full-width-at-half-maximum (FWHM), 66, 70, 72, 75, 219 gain bandwidth, 103, 197 spectrum, 8, 67, 68, 85, 275, 279 temperature dependence of, 275, 279 gain coefficient, 37, 246 gain medium, 67 Gaussian lineshape, 207, 208, 238, 248 group velocity, 27, 100, 141, 202, 210, 211 Hamiltonian, 69, 270 heat capacity, 176 heterogeneous integration, 235 high speed optical communication, 245 hybrid mode, 32 hyperbolic metamaterial, 98, 264 III-V/Si, 235, 256 inorganic semiconductor, 8 Internet of Things, 231 inversionless laser exciton-polariton laser, 214 Langevin, 255, 271 large-signal modulation, 253 lasing threshold, 2, 14, 18 L-I curve, 161, 162, 164, 250

lifetime Auger recombination, 114, 116, 289 carrier, 68, 288 intraband, 280 non-radiative recombination, 14, 16, 114, 116, 141 photon, 16, 17, 20 spontaneous emission, 83, 99 surface recombination, 114, 116, 141, 195, 288 light emitting device, 14 light-light curve, 18, 97, 116 S-shape, 108, 211 light-matter interaction, 22, 132 lineshape broadening of, 67, 72, 280 cavity, 67 Lorentzian, 70, 72, 75, 111 LL curve. See light-light curve localized surface plasmon (LSP), 93 longitudinal mode, 130, 137, 138, 258 long-range SPP, 96 lossless propagation, 5, 36, 38, 98, 265 lower polariton, 216, 224 magnetic-dipole-like mode, 122 material dispersion, 18, 100, 146 matrix element, 84 dipole, 70, 71, 273 momentum, 67 transition, 275 membrane, 1, 177, 178, 269 metal sputtering, 29, 53, 58 metal-cladding, 12, 59, 146, 165, 236, 240, 267 metallo-dielectric, 42, 56, 114 metal-dielectric interface, 5, 32 metal-dielectric-metal (MDM), 28 metal-insulator-metal. See metal-dielectric-metal metal-semiconductor-metal. See metaldielectric-metal micro-photoluminescence, 59 microscale heat-transfer, 176 laser, 19, 256 microscope objective, 60 mirofluidic and/or optofluidic, 9 modal dispersion, 203, 207, 210, 211 modal gain, 8, 11, 12, 13, 29, 171 mode spacing, 100 mode-gain overlap, 3, 8, 73 modulation bandwidth, 1, 245, 254, 255 modulation speed, 73, 118, 231, 245–255 monochromator, 61, 224, 258 MQW gain, 8, 51, 72, 90, 133, 138, 143, 145, 249, 250 multi-physics design, 146, 168, 200 nano-antenna, 31, 35 nanoLED, 245, 246, 249

Index

323

nanowire, 22 near-infrared, 27, 50, 93, 124, 205 noble metal, 118, 265 normalized mode volume, 249 numerical aperture, 117, 125

radiation loss, 6, 12, 32, 38, 73, 80, 101, 120, 128 Raman laser, 54, 234 reflection coefficient, 146, 157 refractive index units, 64 resistor-capacitor, 1

optical communication, 37, 56, 65, 94, 269 optical interconnects, 132, 166, 232 organic semiconductor, 9

scalability, 101, 231, 235 scanning electron microscope, 58 Scanning Transmission Electron Microscopy (STEM), 25 sidewall tilt, 150, 169, 173, 291, 293 silicon dioxide, 26 silicon laser, 263 silicon photonics, 56, 233, 256 small-signal analysis, 210, 252 small-signal modulation, vii, 245 SPASER, 34 spectral density, 144, 212, 254, 255 spectrometer, 61, 224 spontaneous emission lifetime, 114. See spontaneous emission rate modification. See Purcell effect rate, 15, 17, 65, 87, 114 spectrum, 84 temperature dependence of, 78–88 spontaneous emission rate enhancement, 65 inhibition, 65, 143 spot size, 60, 235 steady-state, 70, 173, 176, 194, 211, 212, 244 strong coupling, 93, 214, 221, 229 Super-luminescent Light Emitting Diode (SLED), 59 surface roughness, 24, 159, 178, 200, 207 surface-emitting, 264 surface-enhanced Raman spectroscopy (SERS), 93

passivation, 177 layer, 147, 178, 189 material, 178 surface, 162, 176 patch antenna, 119, 120 PECVD, 51, 160, 177, 178 photoluminescence, 62 photon density, 17, 18, 245, 253 photonic bandgap, 21, 117 photonic crystal defect laser, 21 photonic crystal laser, 21, 177, 267 broad-area cavity, 21 defect cavity, 21 photonic integrated circuit, 9, 259 photonic mode, 28, 34, 36, 117 photoresist planarization, 160 planar integration, 135 plasma frequency, 31, 81, 98, 119, 203 plasma-enhanced ALD, 178 plasmonic mode, 6, 93, 97, 117 polaritons, 216, 217, 221, 225, 228 population inversion, 10, 97, 214, 215, 224 power consumption, 2, 132, 133, 231, 232, 245, 252, 261, 263, 267 power-to-bandwidth ratio (PBR), 252 Poynting vector, 96, 209, 212, 213 propagation constant, 5, 28, 33, 43, 96, 204, 238, 257 propagation loss, 5, 11, 91, 118, 260 Purcell effect, 1, 15, 18, 65–67, 69–78, 98, 246 Purcell factor, 15, 65, 71–77, 78 spontaneous emission factor, 77 temperature dependence of, 78–80 Q factor, 17, 70, 219, 264 QD gain, 8, 21, 247, 251, 253 quantization volume, 270 quantum efficiency, 14, 139, 261 external, 14 internal, 14 quantum yield quantum efficiency, 14 quasi-Fermi level, 68, 142 electron, 134 hole, 134

TE polarization, 100, 121, 125, 238, 279 temperature effect cavity modes, 80–84 insensitivity of spontaneous emission factor, 88–90 spontaneous emission, 80–84 spontaneous emission factor, 78–88 temporal coherence, 33 thermal conductivity, 169, 174, 176, 187, 200 thermal equilibrium, 141, 216, 223, 275 thermal reservoir, 69, 74, 272 thresholdless, 1, 18, 32, 65, 264 TM polarization, 98, 120, 279 total internal reflection, 32 total optical energy confinement factor. See energy confinement factor transfer matrix method, 39, 98, 204, 219, 220 transient, 93, 107, 173, 211 transmission coefficient, 146 transparent conducting oxides (TCO), 265, 266 indium tin oxide (ITO), 205

324

Index

transparent propagation, 5, 94 transverse electromagnetic (TEM), 96, 102, 103, 109 transverse mode, 21 two-level system, 66, 270, 272 undercut, 130, 135, 149, 169 ratio, 153, 169, 192 upper polariton, 216, 220 vacuum wavelength, 41, 211 Vertical cavity surface-emitting laser (VCSEL), 19 volume plasmon-polariton (VPP). See Bloch plasmon-polariton (BPP)

wafer bonding, 54, 235 plasma-assisted, 235, 257 wall-plug efficiency, 14, 235 waveguide coupling, 235, 239, 242, 244, 259 wavelength division multiplexing, 133, 232 weak coupling regime, 74, 79, 215, 219, 220, 230, 272, 273 wet etch, 123, 152, 160, 178 whispering gallery mode, 31, 97, 123, 157, 268 zero temperature condition, 69