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Modeling of the transistor vertical cavity surface emitting laser Faraji, Behnam 2011

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Modeling of the Transistor Vertical Cavity Surface Emitting Laser by Behnam Faraji  B.Sc., Sharif University of Technology, 2002 M.Sc., Sharif University of Technology, 2004  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Electrical and Computer Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) May 2011 c Behnam Faraji 2011  Abstract The direct modulation of semiconductor lasers has many applications in data transmission. However, due to the frequency response it has been challenging to use directly modulated lasers for high speed digital transmission at bit-rates above 10 Gbps. With this in mind, designing a laser with a large modulation bandwidth to be used in high data-rate optical links is very important. Transistor lasers (TLs) are a potential candidate for this purpose. Based on these motivations, the main focus of this PhD research is on understanding the physics of the TL and predicting its performance. A detailed model that correctly incorporates the transistor effects on laser dynamics did not exist. The previous models do not differentiate between the bulk carriers and the quantum well (QW) carriers in the rate equations, do not include the effects of the capture and escape lifetimes in the QW, and significantly overestimate the bandwidth. To account for these phenomena, a model has been developed to study the dynamics of the TL. The model is based on the continuity equation in the separate confinement hetero-structure region of the conventional laser and the base region of the TL. It uses the quantum mechanical escape and capture of carriers in the quantum well region and the laser rate equations to ii  Abstract model the laser operation. The model has been used to gain insight into the conventional separate confinement hetero-structure lasers, and the results of the model have been compared with the experimental results obtained for 850 nm vertical cavity surface emitting lasers (VCSELs). Analytical expressions have been derived for DC and AC parameters of the TL operating in common-base and common-emitter configurations. It has been shown that the TL operating in the common-emitter configuration has a similar modulation bandwidth as a conventional laser (∼ 20 GHz). The common-base configuration, on the other hand, has a very large small-signal modulation bandwidth (> 40 GHz) due to bandwidth equalization in the TL. The largesignal performance of the TL has been studied. Finally, it has been shown that the common-emitter configuration with feedback has improved bandwidth by a factor of 1.5 in high bias currents.  iii  Preface Some parts of this thesis are based on several manuscripts, resulting from collaboration between multiple researchers. Some parts of the Chapter 3 appeared in: B. Faraji, W. Shi, D. L. Pulfrey, L. Chrostowski, “Analytical modeling of the transistor laser,” IEEE Journal of Selected Topics in Quantum Electronics, 15(3), 594 - 603, 2009. B. Faraji, W. Shi, D. L. Pulfrey, L. Chrostowski, “Common-emitter and common-base small-signal operation of the transistor laser,” Applied Physics Letters, 93(14), 2008. B. Faraji, D. L. Pulfrey, L. Chrostowski, “Small-signal modeling of the transistor laser including the quantum capture and escape lifetimes,” Applied Physics Letters, 93(10), 2008. These articles were co-authored with Wei Shi, Prof. David Pulfrey, and Prof. Lukas Chrostowski. The author’s contributions in these publications were developing the main idea, numerical simulation, and writing the manuscript. Wei Shi contributed to the publications through discussions and editing the manuscripts. Prof. Pulfrey and Prof. Chrostowski helped with their numerous suggestions in the course of the devolvement of the model. They also assisted by editing the manuscript. iv  Preface A version of Section 3.4 appeared in: B. Faraji, N. A. F. Jaeger, L. Chrostowski, “Modelling the effect of the feedback on the small signal modulation of the transistor laser,” Photonics Society Annual Meeting, Denver, Colorado, US, 2010. This publication was co-authored with Prof. Nicolas Jaeger and Prof. Lukas Chrostowski. The author’s contributions in this publication were developing the main idea, mathematical analysis, numerical simulation, and writing the manuscript. Prof. Jaeger and Prof. Chrostowski gave the main idea and assisted in the manuscript writing. A version of section 3.5 has appeared in: W. Shi, B. Faraji, M. Greenberg, J. Berggren, Y. Xiang, M. Hammar, M. Lestrade, Z. Li, S. Li, L. Chrostowski, “Design and modeling of a transistor vertical-cavity surface-emitting laser,” Optical and Quantum Electronics, 1 - 8, 2011 (Invited). This work was co-authored with Wei Shi and Prof. Lukas Chrostowski. The author’s contributions in this publication were developing analytical and numerical model, simulation, and editing the manuscript. Wei Shi did the numerical simulation with Crosslight and wrote the manuscript. Prof. Chrostowski initiated the idea to compare the two simulations and parameter extraction from the numerical simulation and using it in the analytical model.  v  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  Abstract Preface  List of Tables  List of Abbreviations  . . . . . . . . . . . . . . . . . . . . . . . . . xiii  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Dedication  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii  1 Introduction . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . 1.1.1 Laser Resonance . . . . . . . . . 1.1.2 Carrier Dynamics . . . . . . . . 1.1.3 Device and Packaging Parasitics 1.1.4 Driver Circuits . . . . . . . . . . 1.2 Transistor Laser . . . . . . . . . . . . . 1.3 Thesis Objective and Chapter Summary 2 Direct Modulation of Semiconductor 2.1 Introduction . . . . . . . . . . . . . 2.2 Rate Equation . . . . . . . . . . . . 2.3 Gain Compression . . . . . . . . . . 2.3.1 Spectral Hole Burning . . . . 2.3.2 Spatial Hole Burning . . . . 2.3.3 Carrier Heating . . . . . . . 2.4 Carrier Capture and Escape . . . .  . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  1 1 8 9 9 10 10 16  Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  . . . . . . . .  19 19 19 28 29 30 30 32 vi  Table of Contents 2.4.1 2.4.2  2.5 2.6  Modeling Using Effective Capture/Escape Lifetimes . Modeling Using Quantum Capture and Escape Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . VCSEL Parasitic Modeling . . . . . . . . . . . . . . . . . . . Model Verification . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Sample Preparation . . . . . . . . . . . . . . . . . . . 2.6.2 DC Measurements . . . . . . . . . . . . . . . . . . . . 2.6.3 AC Characteristics . . . . . . . . . . . . . . . . . . .  3 Transistor Laser . . . . . . . . . . . . . . . . . . . . . . . 3.1 Bipolar Junction Transistor . . . . . . . . . . . . . . . 3.2 Transistor Laser Modeling . . . . . . . . . . . . . . . 3.2.1 General Consideration . . . . . . . . . . . . . 3.3 DC Analysis . . . . . . . . . . . . . . . . . . . . . . . 3.4 AC Analysis . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Common-Emitter Configuration . . . . . . . . 3.4.2 Common-Base Configuration . . . . . . . . . . 3.4.3 3rd Order Modulation Response Approximation 3.5 Feedback in TL . . . . . . . . . . . . . . . . . . . . . 3.6 Model Verification . . . . . . . . . . . . . . . . . . . . 3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Trade-off between RF Gain and Bandwidth . . 3.7.2 Modeling the Multi-Quantum Well (MQW) TL 3.7.3 Limitation of the Diffusion Model . . . . . . .  . . . . . . . .  4 Large-Signal Analysis . . 4.1 Introduction . . . . . . 4.2 Large-Signal Modulation 4.3 FM Analysis . . . . . . 4.4 Discussion . . . . . . .  . . . . . . . . . . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  34 40 52 53 53 55 56  .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  62 62 65 68 72 78 82 85 95 97 102 106 106 109 110  . . . . .  . . . . .  . . . . .  . . . . .  112 112 114 123 124  . . . .  5 Conclusion and Future Works . . . . . . . . . . . . . . . . . . 126 5.1 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . 126 5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131  vii  Table of Contents  Appendices A Derivation of the Kirchhoff ’s Current Law in the Transistor Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 B Derivation of Charge Conservation for the Virtual States in the QW Region . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 C Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 C.1 Peer-Reviewed Journal Publications . . . . . . . . . . . . . . 141 C.2 Peer-Reviewed Conference Papers . . . . . . . . . . . . . . . 142  viii  List of Tables 1.1  Optical communication links. . . . . . . . . . . . . . . . . . .  2.1  Simulation laser parameters. The values are chosen for a typical 980 nm VCSEL. . . . . . . . . . . . . . . . . . . . . . Typical simulation laser parameters. The values are chosen for typical 980 nm VCSELs. . . . . . . . . . . . . . . . . . . . Extracted values for the laser modulation transfer function. .  2.2 2.3 3.1 3.2  3.3  6 23 49 59  Values of the parameters used in the simulations. The values are chosen for a 980 nm TVCSEL. . . . . . . . . . . . . . . . 72 Equivalent 3rd order model parameter values found by curvefitting. The two numbers in each row correspond to values for IB = {2IB,th , 3IB,th }. . . . . . . . . . . . . . . . . . . . . . 97 Values of the parameters. The first column shows the parameters, the second column is the value obtained by comparing our model result with the numerical simulation. The third column shows the values we used in our model. The values are chosen for a 980 nm TVCSEL. . . . . . . . . . . . . . . . 106  ix  List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16  Schematic of the VCSEL. . . . . . . . . . . . . . . . . . . . . Fiber optic analog link. . . . . . . . . . . . . . . . . . . . . . Laser direct modulation response limitation. . . . . . . . . . . Simple schematic of the TL . . . . . . . . . . . . . . . . . . . Carrier distribution in the SCH region of a conventional laser and base region of the TL . . . . . . . . . . . . . . . . . . . . Schematics of the different configurations of the TL. . . . . . TVCSEL diagram. . . . . . . . . . . . . . . . . . . . . . . . .  3 5 7 11  Direct modulation response of 1550 nm VCSEL. . . . . . . . Direct modulation response of the VCSEL using 1-level rate equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Different carrier dynamics involved in the laser operation. . . Direct modulation response of the VCSEL using 1-level rate equations with gain compression. . . . . . . . . . . . . . . . . Carrier transport in an SCH laser. . . . . . . . . . . . . . . . Carrier transport model. 2-level system for carriers. . . . . . Direct modulation response of the VCSEL using 2-level rate equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carrier transport model. 3-level system for carriers. . . . . . Semi-classical model for the carrier dynamics. . . . . . . . . . Flowchart of the simulation. . . . . . . . . . . . . . . . . . . . Carrier distribution at the different bias currents. . . . . . . . Direct modulation response of the VCSEL using semi-classical model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Small-signal modulation bandwidth variation. . . . . . . . . . High speed 850 nm VCSEL in different biasing regimes. . . . DC measurement of a 850 nm VCSEL. . . . . . . . . . . . . . AC measurement experimental setup. . . . . . . . . . . . . .  20  13 14 17  24 29 32 33 35 36 38 41 48 50 51 51 54 56 57  x  List of Figures 2.17 Measured small-signal modulation responses at different current injection levels. The low-frequency fluctuations are mainly from the detector used in measurements. . . . . . . . . . . . . 2.18 Parasitic estimation method. . . . . . . . . . . . . . . . . . . 2.19 Parasitic estimation method. . . . . . . . . . . . . . . . . . . 3.1 3.2 3.3 3.4 3.5 3.6  3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18  58 60 61  An n-p-n transistor. . . . . . . . . . . . . . . . . . . . . . . . 63 Carrier diffusion and quantum capture in the QW, and the conduction band energy diagram of the base region. . . . . . 66 Minority carrier density distribution. . . . . . . . . . . . . . . 76 DC and AC current gain of the transistor and LI curve of the laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Common-emitter configuration of an n-p-n TL. . . . . . . . . 83 Normalized small-signal modulation response of the transistor laser in common-emitter configuration with respect to the different device currents. . . . . . . . . . . . . . . . . . . . . . 84 Common-base configuration of an n-p-n TL. . . . . . . . . . . 85 Small-signal modulation response of the transistor laser in common-emitter and common-base configurations. . . . . . . 86 Intrinsic modulation response of the TL. . . . . . . . . . . . . 88 Current transport factor in common-emitter and commonbase configurations. . . . . . . . . . . . . . . . . . . . . . . . . 89 Small-signal current gain of the transistor. . . . . . . . . . . . 90 Transfer function for the small-signal modulation of a transistor laser with reduced effect of the transistor. . . . . . . . . 91 Bandwidth variation of the transistor laser and intrinsic response, versus DC bias current. . . . . . . . . . . . . . . . . . 92 Energy band diagram of the common-emitter and commonbase configurations. . . . . . . . . . . . . . . . . . . . . . . . . 93 Curve fitting to the small-signal modulation of the transistor laser in common-emitter and common-base configurations . . 96 Simulation circuit. . . . . . . . . . . . . . . . . . . . . . . . . 98 Small-signal modulation transfer function of the TL with feedback. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 PICS3D simulation and measurement data. LIV curves of a MQW In0.17 Ga0.83 As / GaAs VCSEL. The photoluminescence data for the simulation and experiments are shown in the inset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103  xi  List of Figures 3.19 Structure of transistor VCSEL. It is an N-p-n InGaP/GaAs HBT with 30 and 24 pairs of AlGaAs/GaAs layers as bottom and top DBRs, respectively. . . . . . . . . . . . . . . . . . . . 3.19 The LI curve and IC vs IB curve. . . . . . . . . . . . . . . . . 3.20 Transfer function of the small-signal modulation of the transistor laser in the common-emitter and common-base configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.21 Cascode configuration. Q2 is the TL while Q1 is the driving transistor acting as a high output resistance current source. . 4.1 4.2 4.3  4.4 4.5 4.6 4.7  104 105  107 108  Turn-on delay of a SCH VCSEL. . . . . . . . . . . . . . . . . 114 Turn-on delay . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Turn-on delay for a TL in common-base configuration. Laser threshold is IB,th = 1 mA. Output optical power and carrier density variations are shown. The emitter bias current at t = 0 increases from zero to IE (0+ ) = 46 mA. This value corresponds to the 2IB,th . . . . . . . . . . . . . . . . . . . . . 117 Eye-diagram of the digital modulation of the TL in the commonemitter and common-base configurations for different bit-rates.119 Eye-diagram of the digital modulation of the TL in the commonemitter and common-base configurations for different bit-rates.121 Eye-diagram of the digital modulation of the TL in the commonemitter and common-base configurations for different bit-rates.122 Large-signal FM analysis of the TL in the common-emitter and common-base configurations. . . . . . . . . . . . . . . . . 125  xii  List of Abbreviations 1D  One Dimensional  2D  Two Dimensional  3D  Three Dimensional  AC  Alternative Current  AOC  Active Optical Cables  BCB  Benzocyclobutene  BJT  Bipolar Junction Transistor  DBR  Distributed Bragg Reflector  DC  Direct Current  DFB  Distributed Feedback  DOS  Density Of States  DWDM  Dense Wavelength-Division Multiplexing  FM  Frequency Modulation  FP  Fabry-Perot  FTTH  Fiber to the Home  GigE  Gigabit Ethernet  HBT  Heterojunction Bipolar Transistor  LO  Longitudinal Optical  NRZ  Non-Return Zero xiii  List of Abbreviations  PDF  Probability Distribution Function  QW  Quantum Well  RF  Radio Frequency  Rx  Receive  SCH  Separate Confinement Heterostructure  SL  Semiconductor Laser  SMSR  Side Mode Suppression Ratio  TL  Transistor Laser  Tx  Transmit  TVCSEL  Transistor Vertical Cavity Surface Emitting Laser  VCSEL  Vertical Cavity Surface Emitting Laser  WDM  Wavelength-Division Multiplexing  xiv  Acknowledgements I would like to thank a great number of wonderful people who made my experience at UBC memorable. First and foremost, I would like to thank my supervisor Professor Lukas Chrostowski for his leadership and support. His passion for scientific problems has taught me a lot. His superb intuition in solving the problems has been a great asset for this project. The open and free environment that Professor Lukas Chrostowski has created in his laboratory provided me a once-in-a-lifetime opportunity to have a hand in different projects and increase my knowledge. He has been a great advisor and a very good friend for me and I will greatly miss him when I leave UBC. I wish to thank Professor David Pulfrey. We had a very close collaboration when I started the modeling of the transistor laser. Professor Pulfrey with his deep knowledge in semiconductor physics and modeling contributed to this project in many situations. I would like to thank Professor Nicolas Jaeger for his very fruitful discussions on modeling and measurements. Professor Nicolas Jaeger has generously provided many optical parts needed in the experiments. In the beginning of my PhD, we collaborated with Professor David Plant’s group at McGill. I would like to thank him for his valuable insights on the various works we did together. I wish to thank Proxv  Acknowledgements fessor Markus-Christian Amann and members of his group for providing us with 1550 nm VCSELs that were used in the injection-locking experiments. I wish to thank my committee members, Professor David Pulfrey, Professor Konrad Walus, Professor Jeff Young, and Professor David Jones for reading the thesis and giving useful feedback. I wish to thank Professor Shahriar Mirabbasi for being the chair of the defense session. I wish to thank my former mentors in Sharif University of Technology: Professor Sima Noghanian and Professor Foroohar Farzaneh who were my M.Sc. supervisors. I would like to thank Professor Kasra Barkeshli (RIP) to whom I attribute my passion for electromagnetic theory. During my PhD, I had the chance to collaborate with brilliant researchers. I wish to extend my especial thanks to the members of the optoelectronic lab. They have helped me academically and non-academically. I would thank Wei Shi, Sahba Talebi Fard, Raha Vafai, Dr. Mark Greenberg, Miguel Guillen, Xu Wang and former members Eric Kim, Yiyi Zeng, and Qing Gu. I would like to thank Alfred Lam, who helped me to learn the optical measurements upon my arrival at UBC. The Iranian community in Vancouver is a very friendly society and I have had very good friends with whom I shared many wonderful memories. I wish to thank Maryam Shahrokni, Farid Molazem, Azadeh Goudarzi, Pooya Jaferian, Nina Rajabi Nasab, Farzad Moghimi, Nasim Massah, Keivan Ronasi, Ahamad Ashouri, Nazanin Shabani, and Hamid Mohammadi. I wish that our friendships last forever. I would like to thank my uncle Vahid who has a very important role in the fact that I chose science as my path in life. I can still taste the sweetness xvi  Acknowledgements of the first science book that he thoughtfully picked for me. That book was my first inspiration, the first magical spark that dragged me towards science and made me step in to the never-ending path of science education for which I consider my PhD education only a beginning of a lifetime of research and experimentation. I would like to thank my wife’s family, Fereidoon Edelkhani, Farideh Kamali, and Nima Edelkhani. I am very proud to be a member of their family and thank them for their generosity, help, and support during the last three years. With their presence in my life I never felt homesick. I wish to thank my family members. My dad, Sirous, and my mom, Shahin, are the best parents one can ask for. Their unconditional love and generosity have been the most precious blessings in my life. Words cannot describe how much I love them. My brother, Mehdi, and my sister, Nastaran, have always been there for me and have kindly helped me to achieve my goals. The sweetest part of my PhD was when I met Sahar (my wife) and we decided to share the rest of our lives together. She has been my best friend and support. I am blessed to have such a wonderful and beautiful wife. I cannot describe how much I appreciate her sacrifices and encouragements. Her love, patience, kindness, and sacrifices were the bright light to this journey. She is the shining star of my entire life.  xvii  Dedication  This work is dedicated to: Sahar, my beautiful and wonderful wife, Sirous and Shahin, my amazing father and mother, Mehdi and Nastaran, my kind brother and sister.  xviii  Chapter 1  Introduction 1.1  Motivation  Lasers came to be during the 1960s. Basically, a laser is an optical oscillator. Similar to its electrical counterpart, it is a feedback system with an amplification mechanism. The feedback is provided through the mirrors, e.g., dielectric-air interface in edge emitter lasers and distributed Bragg reflector (DBR) in vertical cavity surface emitting lasers (VCSEL). Two mirrors along with the material between them compromise an optical resonator [1]. The resonator creates a frequency-selection mechanism. Two conditions must be satisfied to have oscillation: • The amplifier gain should be higher than the loss in the feedback system • The total phase shift in a single round trip must be a multiple of 2π The first condition is achieved by population inversion via pumping the material and the second condition is satisfied by the optical resonator. The useful output is extracted by coupling a portion of the optical power out of the oscillator.  1  1.1. Motivation On 16 May 1960 the first laser came out of Hughes Research Laboratories in Malibu, California, USA [2]. The first laser was based on ruby, which was pumped by a pulsed photographic flash-lamp. From that time there has been extensive research and investment put into lasers. Lasers have gone through a myriad of changes and innovations and many new applications have been developed, e.g., optical communication, chip interconnect. Direct energy gap materials, e.g., GaAs, InP were obvious candidates to generate light. Semiconductor lasers (SLs) began in 1962 with III-V alloys (GaAs and alloy GaAsP), developed by four research groups [3–6]. In the beginning they were pulse-operated simple pn junctions; they slowly developed into double hetero-structures during the 1970, operating continuously at room temperature. In 1977 quantum wells (QWs) were used in SLs to enhance the density of the states and laser performances [7]. The important properties of the SLs are: • Small size, e.g., cavity length of distributed feedback (DFB) lasers : ∼ 200 μm and of VCSELs: 1 - 2 μm). • Low power consumption (VCSELs: few mW, DFB: few 10 mW). • Direct modulation of the output light. • Semiconductor based fabrication. • Wide range of wavelengths and optical powers (from 0.4 to 10 μm). VCSELs are a class of semiconductor lasers that confine the light through use of the DBRs. Figure 1.1 shows a schematic of a VCSEL. DBRs are fabricated by stacking lattice-matched low and high refractive index layers with the 2  1.1. Motivation proper widths. The gain regions are provided by using QWs. To increase the confinement of the carriers, a separate confinement heterostructure (SCH) is used. For VCSELs based on the Alx Ga1−x As material system one or more oxidation layers are used to confine the pump current to a small area and reduce the threshold current. DBRs are highly reflective, so the cavity is small (1 - 2 μm) and VCSELs essentially are single longitudinal mode lasers and have a low threshold current (< 1 mA for mesa with raduis of ∼ 10 μm)  Direction of the Light  P Contact  P-DBR mirror Oxide Layer  QWs and SCH Layer N-DBR mirror  N Contact  Substrate  Figure 1.1: Schematic of the VCSEL. P-doped and n-doped DBRs are used as the mirrors and oxide layer is used to confine the light to small area to reduce the threshold of the laser. Current is pumped from the top contact and QWs are used as the active region. SCH layer is used to confine the electrical carriers.  The material system used in the VCSEL design depends on the application and consequently on wavelength. For short-range applications (850 nm and 980 nm) DBRs are fabricated by using Alx Ga1−x As material system.  3  1.1. Motivation Oxide confinement is based on the Al-based layers. In VCSELs, because of the top-emitting geometry, fabrication costs are reduced. To give a number, a VCSEL in a computer mouse costs just 10 cents to be fabricated [8]. VCSELs have found many applications in the field of optical communication, especially in short-range data transmission. For example, a 12-fiber ribbon cable and a 1x12 array of VCSELs each sending data at 10 Gbps provides 100-Gigabit Ethernet (100 GigE). Such an example of space division multiplexing is just coming onto the market [8]. An emerging application for VCSELs is active optical cables (AOCs). Datacom companies are making networking easier for data-center companies by attaching optical transceivers to the ends of optical fibers. AOCs use the same pin configuration as the traditional copper cables but carry the signal over an optical fiber. AOCs are much lighter than metal wires. A typical AOC operates at 850 nm and uses VCSELs at both ends, each working at 5 Gbps or higher. IBM also used VCSELs in the Terabus project [9]. Terabus was based on a chip-like optoelectronic packaging structure assembled directly onto a card with integrated waveguides. Each Tx or Rx module consists of a 4 x 12 array of VCSELs or photodiodes that are flip-chip bonded to the driver and receiver IC array. Transmitter and receiver operation was demonstrated up to 14 and 20 Gbps per channel. Figure 1.2 shows a typical optical communication link. It includes a transmitter, transmission medium and a receiver. The goal of an optical link is to send the RF signal so that at the receiver end a replica of the 4  1.1. Motivation    
               
             
      Figure 1.2: Fiber optic analog link. An electrical signal with a center frequency of fRF is transmitted by an optical transmitter with a center frequency of fOptical . An optical fiber is the communication channel. In the receiver side a high speed photo-detector recovers the electrical signal. signal is reproduced reliably. The details of the optical link (e.g., laser type, modulation scheme, fiber length, and multiplexing method) depend on the application and they vary widely according to the system requirements. Table 1.1 summarizes the most important properties of optical links. Typically, the fiber, optical amplifiers, photo-detector, and electronic amplifiers have very good performance in terms of linearity. It is the optical transmitter that usually limits the performance of the link. The optical transmitter includes a semiconductor laser at the desired wavelength. In general there are two ways to modulate laser output power: external modulation and direct modulation. External modulation is achieved by using optical modulators. As it offers the highest performance, it is currently prevalent in most analog links and in digital links with bit-rates above 10 Gbps. Very high bandwidths, > 100 GHz, can be achieved with external modulators [10]. However, there are several disadvantages: the optical modulator component is bulky, expensive and consumes much RF power. Furthermore external modulators have an optical loss, typically 3-6 dB, 5  Table 1.1: Optical communication links.  Network Type  Long-Haul  Metro  Access (e.g., FTTH)  Interconnect  Limited to 20 km by International Telecommunication Union standard  < 100m  > 100 km  > 10 km  Laser type  mainly DFB  DFB, VCSEL  Downstream: DFB, Upstream: DFB or FP  1550 nm (Cband), 1565 to 1625 nm (L-band)  1310 nm, 1550 nm  Downstream: 1490 nm, and/or 1550 nm, Upstream: 1310 nm  850 nm, 1310 nm  Modulation  Direct or external  Direct  Direct  Direct  Speed  10 Gbps  10 Gbps  Downstream: <2.5 Gbps, Upstream: <1.24 Gbps  10 GigE, 40 GigE, and 100 GigE (expected) standards  Multiplexing  WDM, DWDM or coarse WDM  Coarse WDM, DWDM  WDM  Dictated by Ethernet protocols and fiber channel  Wavelength  VCSEL  1.1. Motivation  Distance  6  1.1. Motivation  Bias-T RF  DC Laser Diode Package Parasitics  Laser Diode Chip Parasitics  Laser Diode  H(f)  H(f)  H(f)  f  f  f  Figure 1.3: Intrinsic and extrinsic limitation to the laser-chip direct modulation response. Laser is biased by using a DC source and RF signal is added by using a bias-T. The overall direct modulation response of a laser is affected by three sources: Laser intrinsic response (right square), laser chip parasitics (middle square)and package parasitics (left square). mainly due to coupling loss. Direct modulation of the laser diode is achieved by modulation of the electrical current which modulates the optical power. This method is easy to implement and has many applications. In general, the direct modulation bandwidth of a laser is limited by extrinsic and intrinsic effects. The intrinsic effects include laser resonance and carrier dynamics and the extrinsic effects include parasitics (from the chip and packaging) and driver circuits (RF source limitation and bias-T). Figure 1.3 shows the contribution of the intrinsic and extrinsic limitation to the laser chip direct modulation response. The limitations to laser direct modulation bandwidth are discussed next.  7  1.1. Motivation  1.1.1  Laser Resonance  Experiments have shown the existence of the resonance frequency, referred to as the relaxation oscillation frequency, which results from the interplay between the optical field and the population inversion. The physics of the relaxation oscillation frequency will be discussed in Chapter 2. A low resonance frequency is the most fundamental limitation on laser bandwidth. By small-signal analysis of the rate equations, the relaxation oscillation frequency can be obtained from [11]:  fRO =  G 0 S0 , τp  1 2π  (1.1)  where G0 is the temporal growth coefficient (cm3 s−1 ), S0 is the photon concentration in the active region (cm−3 ), e.g., a QW, and τP is the photon lifetime (s) and is given by:  τP =  n c 1 α− ln(R1 R2 ) 2L  ,  (1.2)  where L is the length of the cavity, R1,2 the reflectivity of the facets, α is the cavity loss (m−1 ), n is the active region group refractive index of refraction, and c is the speed of light in the vacuum. The modulation bandwidth of the laser is widely accepted to be equal to 1.55fR for low damping values [11]. Equation (1.1) gives three ways to increase the laser bandwidth; i.e., increasing optical gain (G0 ), increasing steady state photon density (S0 ), and decreasing photon lifetime (τP ). 8  1.1. Motivation By engineering the laser structure and material properties and maximizing the fR in equation (1.1), laser bandwidth can be increased. However, there is a trade off in optimizing the parameters of the equation (1.1). For example, decreasing the length of cavity (L) will reduce the photon lifetime (τP ), however, such a laser has to be driven at a higher current densities and the thermal effects due to excessive heating will reduce the gain hence limiting the modulation bandwidth [12].  1.1.2  Carrier Dynamics  For a high-gain structure such as a strained QW in addition to the parameters of the equation (1.1), carrier transport can significantly affect the small-signal response of the laser and its bandwidth [13]. Carrier transport puts an upper limit on the laser bandwidth. The origin of these effects is in the interaction of the carriers with each other and the lattice atoms. Diffusion, tunneling, thermionic emission, and transition from continuum energy states to bound energy states and gateway states (e.g., from bulk material to QW) [14] are the main effects that limit the laser bandwidth [13]. To attempt to reduce the carrier transport effect on laser bandwidth the physics of carrier dynamics should be known. These effects are modeled by using lifetime constants and the laser rate equations are changed accordingly [13].  1.1.3  Device and Packaging Parasitics  Device and packaging parasitics are the extrinsic contribution to the laser modulation response. The parasitic response results principally from bond wire inductance, contact resistance, the capacitance associated with the 9  1.2. Transistor Laser chip, and depletion-layer capacitance [15]. These parasitic elements introduce a low-pass filter between the modulation current and laser diode. In VCSELs, the parasitics are modeled by a first order RC circuit, where the resistance is determined by the DBR layers’ resistance, contact resistance, and sheet resistance (resistance between the active region and bottom contact); capacitance is determined by the junction capacitance and oxide capacitance. For typical VCSELs the 3 dB bandwidth of the parasitics is in the range of 3 - 6 GHz [16].  1.1.4  Driver Circuits  The driver circuit imposes some limitations on laser modulation bandwidth. The driver circuit delivers tens of milliamperes of current. The scaled technologies impose lower supply voltages making it complex to design broadband and high power driver circuits for laser diodes used in optical communication. A 60 GHz transceiver on 0.13 μm CMOS technology was designed in [17]. In [18], the authors have implemented driver circuits that can supply 11 mA of modulation current at data rates up to 20 Gbps for a 4×12 array of VCSELs.  1.2  Transistor Laser  The first integration of the laser and the transistor was reported in [19]. In that work, a two-state device was fabricated and according to the voltage polarity on the junction the device could toggle between the emission state and non-emission state. The device had an heterojunction bipolar transistor  10  1.2. Transistor Laser (HBT) structure to confine the holes in the base region while operating in the emission state. The intensity of the output light could be controlled by the base current. More recently, a TL operating at room temperature was fabricated and demonstrated by using a QW in the base of the HBT [20, 21]. The transistor laser (TL) is a three-contact device that exploits the transistor to enhance some of the characteristics of the laser diode. A very simple schematic of a TL is shown in Figure 1.4. Figure 1.4 (a) shows the important elements of a TL. It has three electrical ports, i.e., an emitter, a base, and a collector, it has one QW that generates the laser light. The light travels in the direction perpendicular to the paper. Laser mirrors are not shown in the figure. The details of the transistor biasing and the junctions behaviors will be discussed in the Chapter 3. Figure 1.4 (b) shows the circuit schematic for the TL. The emitter-base junction is shown with an arrow and the direction of the arrow is from the p-doped side to the n-doped side.  IB  IE  IC  IC Laser Light  QW  Base  Collector Laser Light  IB  Emitter  Base (a)  Collector  Emitter IE  (b)  Figure 1.4: Simple schematic of the TL. a) Important elements in the TL. It consists of an emitter, base, and collector of a transistor and a QW inside the base. The QW acts as an optical collector and converts the carriers to photons. b) Circuit schematic of the TL. The operation of the TL is based on the operation of the transistor.  11  1.2. Transistor Laser Part of the carriers (electrons or holes for the n-p-n or p-n-p transistors, respectively) that are injected from the emitter to the base generate light and part of them are swept to the collector and contribute to the collector current. As the base-collector junction is considered as an alternative path for removing the carriers from the active region of the laser diode, the carriers effective lifetime is reduced and the TL has attracted attention as a solution to the limited modulation bandwidth of the laser diode. Figure 1.5 shows the carrier distributions in the SCH region of the regular laser diode and base region of the TL. In Figure 1.5 (a), as the carriers are injected into the active region, because of the energy barriers they will be confined in the SCH region and the mechanisms to remove the carriers are recombination (radiative and non-radiative) and stimulated emission. In Figure 1.5 (b), the reverse-biased collector-based junction in the TL forces the minority carrier concentration to be (almost) zero at the collector-base boundary. The collector current originates from this tilted distribution [22] and creates another path for the carriers to get depleted from the base region. In Chapter 3, we will see that this effect changes the dynamics of the laser greatly. Because of the removal of the carriers from the base region through the base-collector junction, the average lifetime of the carriers is reduced. This reduction of the lifetime causes new effects such as RF gain and modulation bandwidth equalization in the common-base and common-emitter configurations [23], and modified carrier dynamics [24] take place in the TL. Because of the stimulated emission of the laser, some of the characteristics of the transistor are changed. These include reduction in the AC and DC current 12  1.2. Transistor Laser  E  Carrier Injection Carrier Distribution  Doped Cladding  SCH  SCH  Doped Cladding  QW x (a)  E  Carrier Injection Carrier Distribution  Emitter-Base Junction  Base  Base  Collector-Base Junction  QW x (b)  Figure 1.5: Carrier spatial distribution in different laser structures. a) Carrier distribution in the SCH region of the conventional laser diode. Because of the energy barrier the carries are confined in the SCH region b) Carrier distribution in the base layer of the TL. Because of the reversed-biased basecollector junction the carriers concentration is very small at the interface of the base-collector. gain, compression in the carrier distribution [24], and compression in the IV characteristics [25]. Small-signal modulation measurements are presented in [26] and a bandwidth of 13.5 GHz is obtained while it is concluded that the intrinsic bandwidth of TL is 44 GHz. Different modulation configurations can be considered for the TL. This is because the transistor has three ports. These are briefly explained in Figure 1.6 where just the AC part of the circuits are shown. In the common13  1.2. Transistor Laser  Laser Light Laser Light  Laser Light  (a)  (b)  (c)  Figure 1.6: Schematics of the different configurations of the TL. These are the AC schematics and the details of the biasing are not shown. a) Commonemitter: the RF signal is applied to the base and the emitter is grounded. b) Common-base: the RF signal is applied to the emitter and the base is grounded. c) Common-collector: the RF signal is applied to the base and the collector is grounded. vo is the output electrical voltage and RL represents the load resistance of the next stage. emitter, Figure 1.6 (a), the emitter is grounded and the RF signal is applied to the base and the electrical output is taken from the collector. Figure 1.6 (b) shows the common-base where the RF signal is applied to the emitter and the base is grounded and the collector gives the electrical output. Figure 1.6 (c) shows the common-collector where the RF signal is applied to the base, the collector is grounded and the electrical output is taken from the emitter. For regular transistors each configuration has its own characteristics and applications and they are discussed in micro-electronic books [27] and they will be reviewed briefly in Chapter 3. In the TL, however, because of the QW the situation is different and we need to study each configuration separately to find out the features. In [28], the authors presented a model based on the charge control  14  1.2. Transistor Laser method and laser rate equations, which predicts a large intrinsic modulation bandwidth in the common-emitter configuration. However the model does not differentiate between the bulk carriers and the QW carriers in the rate equation and does not include the effects of the capture and escape lifetimes in the QW, and significantly overestimates the bandwidth, i.e., a 3 dB bandwidth of 70 GHz for an edge emitting TL. Recently, in [29], a model based on the charge control model for the transient analysis of the bipolar transistor laser has been developed to describe the dynamics of electron, photon and charge densities. The model is based on the integration of the bulk carrier densities with the QW carrier densities through the quantum capture with a constant lifetime. It predicted 55 GHz bandwidth for the common-emitter configuration and the authors also report the large-signal modulation performance of the TL. This model does not give the spatial distribution of the carriers and does not consider the QW carrier escape and capture processes. New formulation is needed to correctly predict the modulation bandwidth observed in experimental devices in the common-emitter configuration, e.g., [26]. In [23], by modeling we compared the optical and electrical performances of the common-base and common-emitter configurations. We showed that the TL modulated in the common-base configuration can have a bandwidth equalization which significantly increases the 3 dB modulation bandwidth and its response can suppress relaxation oscillations which is important in distortion-free modulation [30]. For TLs there are two possibilities: edge-emitting TLs and verticalemitting TLs. In particular VCSELs are very interesting because of their 15  1.3. Thesis Objective and Chapter Summary importance in the next generation of optical communication systems, where many very low-cost transmitters will be needed. These transmitters could be constructed from our proposed high-speed transistor VCSELs. Figure 1.7 shows the TVCSEL structure and the details of the device we have designed. Figure 1.7 (a) shows the cross-sectional view of the TVCSEL. It consists of three contacts: emitter, base, and collector. The base region is between the top and bottom DBRs and emitter and collector contacts are on the top and bottom DBRs, respectively (intra-cavity contacts). Figure 1.7 (b) shows the top view of the TVCSEL. The contacts are annular regions surrounding the VCSEL active region.  1.3  Thesis Objective and Chapter Summary  This work studies the optical and electrical characteristics of the TL. The objective of the thesis is to show that the TL has new features which may be useful for future applications in optical communications. To achieve the objectives the following contributions have been made: • A novel analytical model has been created. This is a 1D model that predicts the necessary characteristics of the device, including optical output power, small-signal and large-signal modulation responses in common-base and common-emitter configurations. • The model has been validated with the experimental data for conventional two-contact VCSELs. • The small-signal modulation response of the TL under different bias 16  1.3. Thesis Objective and Chapter Summary Light Direction Emitter Contact Top DBR Emitter Layer  Oxide Layer Base Contact  Base Layer Active Region Collector Contact  Collector Layer  Bottom DBR  (a) Emitter Contact  Base Contact Optical Aperture Collector Contact  (b)  Figure 1.7: TVCSEL structure a) Cross-sectional view of the TVCSEL. Active region is inside the base layer of the transistor and emitter and collector contacts are on the top and bottom DBRs, respectively. b) Top view of the TVCSEL. The contacts are annular regions. configurations has been studied and comparisons have been made with the responses of regular laser diodes. • The large-signal modulation response of the device has been studied by numerically solving the model. • The model was integrated with external circuits, and the effect of electronic feedback has been studied. This is a new application for the TL, which it can be described by our model.  17  1.3. Thesis Objective and Chapter Summary The organization of the thesis is as follows: In Chapter 2, the rate equations of the laser are reviewed and it is shown how to obtain the modulation response of the laser diode analytically and numerically. The concept of the gain compression and its origins are discussed and the modulation response changes are simulated in the presence of the gain compression. The quantum capture and escape processes are explained and it is shown how to integrate them in the model, and their effects on the modulation responses are simulated. A verification of the model based on a comparison with the experimental data is presented at the end of the chapter. In Chapter 3, the details of the analytical modeling for the DC and smallsignal modulation of a TL are presented. The limitations and possibility for the generalization of the model are explained. The concept of RF gain and optical bandwidth equalization for the TL is presented for the first time. We also show that the small-signal modulation response of the TL in both the common-emitter and the common-base configurations can be represented by a third order transfer function. In Chapter 4, large-signal modulation of the TL in the common-emitter and the common-base configurations is studied. The turn-on delay of the device is investigated and eye-diagrams for different bit-rates are shown. Frequency chirping analysis is presented. In Chapter 5 the thesis is concluded by a brief summary and a discussion which shows the contributions of this thesis in the bigger picture. Future research directions are discussed.  18  Chapter 2  Direct Modulation of Semiconductor Lasers 2.1  Introduction  The semiconductor laser can be modulated directly by modulating the external current. This is very important, as other type of lasers cannot be modulated directly. In this section we review the theory behind laser dynamics and develop the necessary modeling. These concepts will be used in the next chapter to model the transistor laser.  2.2  Rate Equation  Figure 2.1 shows the measured response of the small-signal direct modulation of a VCSEL. This VCSEL is designed for 1550 nm and its threshold is 0.66 mA [31]. The modulation response can be modeled by the rate equations that were introduced in [12] to study the bulk and QW lasers. The rate equations are the bookkeeping of the supply, annihilation, and creation of the electrons and photons inside the laser cavity [12]. In the  19  2.2. Rate Equation  −40  −60 RF Response (dB)  −60 RF Response (dB)  −40  I=1.2Ith I=2Ith I=3Ith I=5Ith  −80  −100  −120  −140 0  −80  −100  I=1.2Ith I=2Ith I=3Ith I=5Ith  −120  5  10  15 f (GHz)  20  25  30  −140 0.01  0.1  (a)  1 f (GHz)  10  30  (b)  Figure 2.1: Transfer function of direct modulation of 1550 nm VCSEL for different bias currents. The resonance peak is obvious in two graphs (2Ith and 3Ith ). The VCSEL had a threshold of 0.66 mA [31]. a) linear scale, b) logarithmic scale. Both scales are used in the literature interchangeably. simplest form the rate equations can be written as [11]: dN I N − vg gS , = − dt qdA τs  (2.1)  S dS + Rsp , = Γvg gS − dt τP  (2.2)  where S is the photon concentration (cm−3 ), N is the carrier density (electron) in the active region, e.g., QWs in the VCSELs (cm−3 ), Γ is the optical confinement factor and is defined as the overlap of the optical field with the active region, g is the material gain (cm−1 ), vg is the group velocity of the c optical field in the active region (cm s−1 ) and equals to where ng is the ng refractive index of the active area, τP is the photon lifetime (s), Rsp is the spontaneous emission that is coupled to the lasing mode (cm−3 s−1 ), I is the electrical pump current (A), q is the electronic charge (C), d is the active  20  2.2. Rate Equation region thickness, e.g., QW width in the VCSEL (nm), and A is the area of the active region (μm2 ). Equations (2.1) and (2.4) are called 1-level rate equations as there is one equation to describe carrier density. In equations (2.1) and (2.4), the recombination processes have been modeled by a single lifetime τS . Generally, recombination processes have wellknown carrier dependency:  R = AN + BN 2 + CN 3 ,  (2.3)  where, N is the carrier density, A is the monomolecular recombination coefficient, B is the bimolecular recombination coefficient, and C is the Auger recombination coefficient which is a function of the valence band structure of the material. In practice, near or above threshold carrier concentration does not vary significantly so an average lifetime can be used in the rate equations which simplifies the analytical expressions [11]. We define: G = vg g ,  (2.4)  as the optical temporal gain (s−1 ). In general, G is a nonlinear function of the carrier and photon densities, but the simplest form is a linear function of the carrier density:  G(N, S) = G0 (N − Ns ) ,  (2.5)  where Ns is the fitting parameter that sometimes is referred as the transparency carrier concentration [11] and G0 is the temporal growth coefficient 21  2.2. Rate Equation and is related to the material gain (g) through:  G0 = v g  ∂g , ∂N  (2.6)  ∂g is called the differential gain. The spontaneous emission term is ∂N given by: N (2.7) Rsp = Γβ , τs  where  where β is the spontaneous emission coupling coefficient. By solving equations (2.1), (2.4), (2.5), and (2.6) numerically (e.g., finite difference), steadystate and frequency responses are obtained. It is important to mention that in the laser diode, active regions are undoped and charge neutrality dictates that the electron density equals the hole density, i.e., N = P . This condition greatly simplifies the modeling and only one type of carrier, e.g., electrons, can be used in the rate equations. Further, the laser works under the forward-biased condition and carrier density levels are about 1018 cm−3 . At this level of injection, unintentional background doping can be ignored [13]. The optical output power is calculated by using:  Po =  αm hc SAd , α m + α i λ τP Γ  (2.8)  where Po is the output optical power measured in mW, αm and αi are mirror loss and internal loss respectively, c is the light velocity, λ is the laser wavelength, and h is the Planck constant. The above equations are solved by using the finite difference method 22  2.2. Rate Equation Parameter Mesa size Carrier lifetime Photon lifetime Group velocity Mirror loss Internal loss Active area thickness Optical confinement factor Fitting parameter Temporal growth coefficient Spontaneous emission coupling coefficient  Unit μm2 ps ps cm/ns cm−1 cm−1 nm cm−3 cm3 s−1 -  Symbol A τS τP vg αm αi d Γ Ns G0 β  Value 9π 500 3 5.13 40 25 12 0.05 1.2 × 1018 3 × 10−6 10−4  Table 2.1: Simulation laser parameters. The values are chosen for a typical 980 nm VCSEL. [32]. The device parameters are shown in Table 2.1. The values are chosen for a typical 980 nm VCSEL. The most important DC characteristic is the laser light-current (LI) curve, which is the optical output power against the bias current. This is shown in Figure 2.2(a). To find the small-signal modulation response of the laser, in the simulations the driving current is taken as:  I(t) = I0 + im sin ωt ,  (2.9)  where I(t) is the total current used in the rate equations, I0 is the bias current, im is the amplitude of the RF current, and ω is the RF angular frequency. To be in the small-signal regime im is assumed to be very small in comparison to I0 . Using equation (2.9) the steady-state optical output  23  2.2. Rate Equation  4  Transfer Function (dB)  20  P (mW)  3  2  1  0  −20  −40  −60  0 0  1  2  3  4  5  I (mA)  −80 0.1  I = 1 mA I = 2 mA I = 3 mA I = 50 mA  −40 dB/dec  1  10  100  f (GHz)  (a)  (b)  Figure 2.2: Rate equations simulations. 1-level equations have been used, no gain compression included. The parameters are chosen for a 980 nm VCSELs. a) LI curve of the VCSEL. Ith = 0.7 mA. b) Modulation response at different bias points. Relaxation oscillation frequency and the damping increase as the bias current increases. The amplitude of the RF current is set to im = 10 μA. 20log10 (pm /im ) is used. power of the laser will be:  Po (t) = P0 + pm sin ωt ,  (2.10)  where Po (t) is the total optical output power, P0 and pm are the DC and the AC part of the optical output power, respectively. The output power small-signal modulation transfer function is defined as:  H(ω) =  pm , im  (2.11)  Figure 2.2(b) shows the small-signal modulation response of the laser. The small-signal modulation curves are for bias points above threshold. To find the transfer function the simulation is done for all RF frequency values  24  2.2. Rate Equation of interest. In Figure 2.2(b), we are using 20log10 (H(ω)). To see the reason, we have to explain how a network analyzer measures the laser transfer function. We are assuming an ideal photo-detector with a responsivity of 1 A/W is present. The output electrical power is calculated as:  2 2 pe = R0 i2e = R0 RD pm ,  (2.12)  where pe is the electrical power, ie is the electrical current, R0 is the impedance of the electrical power meter, RD is the photo-detector responsivity, and pm is the optical power as in equation (2.10). If the injected RF power, used to modulate the device is pin , what a network analyzer shows is:  HN A (ω) = 10 log10  pe pin  ,  (2.13)  where HN A (ω) is the transfer function from the network analyzer in the logarithmic scale. By using equations (2.11) and (2.12) it is easy to show that:  10 log10  pe pin  ∝ 10 log10  p2m i2m  = 20 log10 (H(ω)) .  (2.14)  In this thesis we use equation (2.11) for the simulation results and equation (2.13) for the measurement results. From Figure 2.2(b) a few interesting behaviors of the laser dynamic can be seen. First, there is a peak in the modulation response. This peak comes from the resonance between the photons and active region carriers. Second, 25  2.2. Rate Equation as the bias current is increased the magnitude of the peak first increases then decreases and at very high bias current the peak almost disappears. Third, the slope of the small-signal modulation response in high frequencies is -40 dB/dec, indicative of a two pole system. It is important to mention that the equations (2.1) and (2.4) predict the general behavior of the modulation response (e.g., slope after resonance, resonance dependance on bias current, ... ) correctly, however, they do not provide precise information on the bandwidth and generally, they overestimate it. The resonance between the photons and the carriers in a modulated laser can be explained by the use of the rate equations. We assume the following forms for I(t), N (t), and S(t):  I(t) = I0 + i(t) ,  (2.15)  N (t) = N0 + n(t) ,  (2.16)  S(t) = S0 + s(t) ,  (2.17)  where I0 , N0 , and S0 are the DC values, i(t), n(t), and s(t) are small-signal time-varying values for the modulation current, carrier density, and photon density, respectively. By putting equations (2.15), (2.16), and (2.17) into equations (2.1) and (2.4) and separating DC and time-varying values we obtain the following rate equations for the time-varying values: dn(t) i(t) n(t) s(t) − − G0 S0 n(t) , = − dt qdA τs ΓτP  (2.18)  26  2.2. Rate Equation ds(t) = ΓG0 S0 n(t) . dt  (2.19)  In writing the equations (2.18) and (2.19), spontaneous emission effect is ignored as above laser threshold the spontaneous emission effect is much less than the stimulated emission effect. From equation (2.19), as n(t) increases and becomes positive, s(t) increases due to an increase in the laser gain. From equation (2.18), as s(t) becomes positive it decreases n(t) by increasing the stimulated emission. As n(t) decreases and becomes negative, it decreases s(t). As s(t) becomes negative it increases n(t). This cycles repeats itself and produces resonance in the laser cavity. To find an analytical expression for the small-signal modulation response of the laser, we assume a sinusoidal form for the time varying signals:  i(t) = Re im ejωt ,  (2.20)  n(t) = Re nm ejωt ,  (2.21)  s(t) = Re sm ejωt ,  (2.22)  where im , nm , and sm are amplitudes of the small-signal modulation current, carrier density, and photon density, respectively. By putting equations (2.20), (2.21), and (2.22) into equations (2.18) and (2.19), analytical form of the small-signal modulation response can be obtained: ΓG0 S0  sm (ω) qdA , = im (ω) −ω 2 + jωγ + ωr2  (2.23)  where ωr is the relaxation oscillation frequency and γ is the damping coef27  2.3. Gain Compression ficient and are given by:  ωr = ωr,0 =  G0 S0 , τP  (2.24)  1 . τs  (2.25)  γ = γ 0 = G 0 S0 +  Considering the effect of the spontaneous emission, i.e., β in equation (2.7), these expressions are changed to:  2 ωr2 = ωr,0 +β  ΓN0 γ0 (G0 + ) , τs S0  γ = γ0 + β  ΓN0 . τs S 0  (2.26)  (2.27)  In general, ωr increases as the photon concentration increases, however the damping increases as well which limits the bandwidth of the modulation response. However, the magnitude of the damping observed in the experiments is larger than what was predicted by the spontaneous emission factor. In order to more accurately predict the frequency response, further details need to be included in the model. Non-linearity in the gain is the next improvement in the laser model.  2.3  Gain Compression  The origins of the gain compression lie in the carrier dynamics. The carrier dynamics can be divided into three main categories. 1) inter-band, 2) intra-band, and 3) diffusion. Inter-band processes change the carriers concentration (e.g., recombination), intra-band processes change the carriers 28  2.3. Gain Compression energy within the band (e.g., conduction band or valence band), and diffusion changes the spatial distribution of the carriers. These processes are shown in Figure 2.3.  (a)  (b)  (c)  Figure 2.3: Different carrier dynamics involved in the laser operation. a) Inter-band process changes the carriers concentration. b) Intra-band process changes the carriers energy. c) Diffusion process change the carriers spatial distribution. Intra-band processes are present in bulk and QW lasers while the interband processes are more important in QW lasers. Diffusion processes change the spatial distribution of the carriers in the SCH region. These processes explain the different mechanisms involved in the gain compression. It is important to mention that laser dynamics depends on the carriers distribution change (both in energy and space) and the time constants of the changes.  2.3.1  Spectral Hole Burning  In spectral hole burning, stimulated emission of the laser rapidly removes the carriers with energies close to the lasing wavelength so the optical gain near the lasing wavelength decreases. This is an inter-band process.  29  2.3. Gain Compression  2.3.2  Spatial Hole Burning  Because of the standing wave in the laser cavity, the optical intensity is not uniform in the longitudinal direction. So the carriers are depleted in the vicinity of the intensity peak. Consequently the optical gain is also decreased at the intensity-peak location. This process is called spatial hole burning. Spatial hole burning happens in the lateral direction as well. The optical intensity is not uniform in the lateral direction so the optical gain decreases near the field-intensity peaks. Spatial hole burning is an inter-band process.  2.3.3  Carrier Heating  The pump and probe experiments [33] showed that there is another mechanism that contributes to the gain compression and that is the carrier heating. The carrier heating falls into the realm of intra-band process and it explains the dependence of the optical gain on the energy of the carriers. The carrier temperature is increased by the free carrier absorption and stimulated emission. Free carrier absorption increases the energy of the carriers and hence increases the carrier temperature. Stimulated emission increases the carrier average temperature by removing the low energy, cool carriers. The heated carriers do not contribute to the optical gain until they are cooled down to the lattice temperature. The cooling mechanism is based on the electron-phonon scattering and has a relaxation time of 0.5-1 ps [34]. In general the dependence of the optical gain profile to the above mentioned processes is a complicated function [35]. By adding a phenomenological coefficient to the rate equations [11, 36] the effects of these processes  30  2.3. Gain Compression on the laser modulation response can be modeled. This coefficient which is called the gain compression coefficient, modifies the gain term, G(N, S) such that it becomes photon-density dependant. In this work we use the following form [11]: G(N, S) = G0 where  N − Ns , 1+ S  (2.28)  is the gain compression factor. Figure 2.4 shows the characteristics  of the same laser that was simulated in Figure 2.2. In these simulations gain compression factor was integrated in the rate equations. Figure 2.4 (a) shows the LI curve of the laser for different values of . Comparing this figure with Figure 2.2 (a) shows that non-zero  decreases the slope of the  LI curve of the laser. Slope decreasing can be explained by noting that with the gain compression factor, the carrier concentration in no longer clamped above threshold and a higher inversion is required to maintain lasing. At a specific bias current the photon concentration will be lower to account for the increase in the carrier concentration. Figure 2.4 (b) shows the smallsignal modulation response of the laser at one bias point for several values. Comparing this figure with Figure 2.2 (b), shows that non-zero  decreases  the damping which limits the bandwidth. The low-frequency RF gain is also affected by the non-zero and it can be explained by the decrease in the slope of the LI curve. The gain compression puts a practical upper limit on the modulation bandwidth for the direct modulation [36]. For InGaAs/GaAs material system that is used for 980 nm VCSEL,  = 3×10−17 cm3 is used  in the simulations. Analytically, using the gain compression factor in the rate equations and  31  2.4. Carrier Capture and Escape  20  4  P (mW)  3  10  ε=0 −17  cm  −16  cm  ε = 3x10 ε = 3x10  3  Transfer Function (dB)  3.5  3  2.5 2 1.5 1  1  2  3  4  5  −40 dB/dec  −20 −30 −40 −50  0.5 0 0  0 −10  ε=0 −17  cm  −16  cm  ε = 3x10 ε = 3x10  −60 0.1  3 3  1  10  I (mA)  f (GHz)  (a)  (b)  50  Figure 2.4: Rate equations simulations. 1-level equations have been used and gain compression is included. a) LI curve of the VCSEL for the different values of . Slope of the LI curve decrease for the large values of . b) Samllsignal modulation response for the different values of . Laser is biased at I0 = 1 mA and the amplitude of the RF current is set to im = 10 μA. As the increases the damping increases and bandwidth of the laser decreases. ignoring the spontaneous emission effect, β, the expressions for ωr and γ are written as: ωr2 =  1 G0 S 0 S0 + , τP 1 + S0 τp τs  (2.29)  γ=  1 G0 S0 S0 + + . τs 1 + S 0 τP  (2.30)  The numerical simulations confirm the analytical expressions.  2.4  Carrier Capture and Escape  It has been found that in SCHs with a QW as the active region, transport effects can significantly change the dynamics of the laser [11]. When QW semiconductor lasers were introduced, it was predicted that their direct modulation bandwidth could exceed 60 GHz due to the increased optical 32  2.4. Carrier Capture and Escape 1  2 3 Vb QW Carriers  QW  SCH Carriers  3 1  2  Figure 2.5: Carrier transport in an SCH laser. Different processes are shown with numbers: (1) diffusion in the SCH region moves the electron and holes in the SCH layer to the QW, (2) diffusion in the QW, and (3) quantum capturing moves the carriers to the sub-bands and quantum escaping takes the carriers out of QW. Effective barrier height, Vb , is also shown. gain they provide [37]. However, experiments showed that the bandwidth enhances only a small amount compared with the bulk semiconductor laser. More experiments carried out on the subject of the spontaneous emission from different parts of the SCH QW semiconductor lasers confirmed that the carrier transport can change the modulation response of the laser [38]. These experiments suggested that the SCH carriers should also be considered in the modeling. The important forms of the carrier transport that we need to consider in the modeling are the diffusion of the carriers in the SCH region, quantum capture and escape processes. Figure 2.5 shows these processes. In this figure the conduction and the valence band of the SCH and QW regions are shown. Carriers diffusion brings the carriers to the QW and in the QW quantum capturing process lowers down the carriers energies and quantum escaping process takes the carriers out of QW. 33  2.4. Carrier Capture and Escape We will explain the physical source of these mechanisms and will show how to integrate them into the rate equations. Experimentally, the measurement process of the carrier dynamics (quantum capture, escape, carrier diffusion) is based on the wavelength-selective optical modulation [15]. By selecting the wavelength of the pumping laser to lie either in the SCH region or QW absorption band it is possible to directly modulate the carriers located in the QW or SCH region [39].  2.4.1  Modeling Using Effective Capture/Escape Lifetimes  There have been different ways to integrate the carrier transport effects into the laser rate equations. The simplest way is to add another equation describing the dynamics of the SCH region carriers. This new equation includes the carrier capture, escape and diffusion lifetimes by using effective capture and escape lifetimes. Figure 2.6 shows the reservoir model. In this model additional time lifetimes are used to describe the loss of the carriers from SCH layer and the gain of the carriers by QW (τc ) and loss of carriers from QW and the gain of carriers by SCH layer (τe ). From the figure, the rate equations can be easily derived [13]: dN1 N2 I N1 N1 − + Γe , = − dt qdsch A τs τc τe  (2.31)  dN2 1 N1 N2 N2 − − − GS , = dt Γe τc τe τs  (2.32)  S N2 dS + Γo β , = Γo GS − dt τP τs  (2.33)  in equations (2.31), (2.32), and (2.33) N1 is the carrier concentration in the 34  2.4. Carrier Capture and Escape  E  I  N1  Doped Cladding  τe SCH  N2 QW  τc  Doped Cladding  SCH x  Figure 2.6: Carrier transport model. 2-level system for carriers. Conduction energy band of an SCH laser is shown. Current is injected to the SCH region via the doped cladding layers. In this model the carriers in the continuum energy states of the SCH and QW areas, N1 , are connected to the carriers in 2D states of the QW, N2 , through the capture lifetime, τc and escape lifetime, τe . In this model the gain compression effects are separated from the capturing and escaping lifetimes. continuum states of the SCH region, N2 is the carrier concentration in 2D states in QW, dsch is the SCH region thickness, Γe = d/dsch is the electrical confinement of the carriers to account for the fact that N1 and N2 are normalized with respect to two different volumes, τS is the recombination lifetime in both SCH and QW regions, τc is the effective capture time, and τe is the effective escape time. This modeling leads to the so-called “2-level rate equations” [40]. Equation (2.31) describes the dynamics of the carriers in the continuum energy states of the SCH and QW regions. The first term is this equation is the carriers injected through the electrical current, second term describes the recombination process, third term is the loss of carriers due to the capturing, and last term is the injection of the carriers to the continuum states from 2D QW states. Equations (2.32) and (2.33) are similar to the equations (2.1) and (2.4) except the electrical current injection 35  2.4. Carrier Capture and Escape 10  Transfer Function (dB)  0 −10 −20 −30 −40  I = 1 mA I = 2 mA I = 3 mA I = 10 mA  −60 dB/dec  −50 −60 0.1  1  10  50  f (GHz)  Figure 2.7: Rate equations simulations. Small-signal modulation transfer response at different bias points. 2-level equations have been used. Carriers effective capture and escape, and gain compression are included ( = 3×10−17 cm3 ). The amplitude of the RF current is set to im = 10 μA term is replaced by: 1 N1 N2 − Γe τc τe  (2.34)  Figure 2.7 shows the small-signal modulation response of the same laser studied in the previous section that includes the carrier transport effects. For the simulation, in equations (2.31), (2.32), and (2.33) τc = 6 ps and τe = 100 ps are used [11]. From the study it has become clear that the damping increases and slope of the curve increases in high frequencies. Using this model an analytical expression for the small-signal modulation can be obtained [11]. It can be shown that the effect of the carrier transport is to introduce a low-pass parasitic-like roll-off which limits the maximum possible modulation bandwidth; in the case of the Figure 2.7, it is approximately at 26 GHz. The small-signal modulation response can be 36  2.4. Carrier Capture and Escape written: Ht (ω) =  2 ωt,r s(ω) 1 = · 2 , i(ω) 1 + jωτc −ω 2 + jωγt + ωt,r  (2.35)  in equation (2.35), the resonance frequency has the following shape:  2 ωtr =  ωr2 , χ  (2.36)  where ωr is defined by equation (2.26) and χ is defined as:  χ=1+  τc /τe τc ≈1+ . 1 + ωτc τe  (2.37)  The added pole due to the carrier transport depends only on capture time. The carrier transport reduces the relaxation oscillation frequency. This modification in the rate equations leads to the better agreement with the experimental data [13]. From this simple model some valuable insights are gained for the highspeed laser design. The transport time across the SCH has to be minimized. Long carrier transport delay in the un-doped regions of the laser, as in the SCH, limits the modulation bandwidth. A narrow SCH is desirable toward this end, but this will increase the internal loss in the case where the cladding loss is dominant. It is worth noting that the effect of the carrier transport on the resonance frequency and damping can be modeled as an effective gain compression factor and can be added to the phenomenological gain compression, , from other sources in this case will be larger than 3×10−17 cm3 that is used in the simulations. In this work we separate the effects of the carrier transport, as it will be critical in formulating the transistor effect 37  2.4. Carrier Capture and Escape on the laser response. In the 2-level rate equations model, SCH carriers and QW carriers occupying the continuum energy states are described by one variable, N1 in equation (2.31). There is a 3-level rate equations model that distinguishes between the carriers in the bulk SCH region with the carriers localized in the QW position through the usage of the virtual or gateway states [41]. These states aid in the conversion of carriers between 3D states above the well and 2D states within the quantum well, through phonon radiation. These states are localized to the quantum well, but occupy energies larger than the band gap of the barrier material. Carriers having energies at or just above the  E  I  IJD IJG  N1  N1  N2 Doped Cladding  IJesc SCH  N3 QW  IJcap  Doped Cladding  SCH x  Figure 2.8: Carrier transport model. 3-level system for carriers. In this model the SCH region carriers, N1 , are connected to the carriers of the 3D virtual states, N2 , through the lifetimes of τD and τG . Carriers of the 3D virtual states, N2 , are connected to the carriers in 2D states of the QW, N3 , through the capture lifetime, τcap and escape lifetime, τesc . barrier material band gap are rapidly depleted (captured by the well) under gain conditions [42]. Figure 2.8 shows the schematic of the 3-level rate equations. In 3-level rate equations the gateway states can be viewed as a temporary storage location for carriers which are supplied from the SCH.  38  2.4. Carrier Capture and Escape Stored charge is lost either by capture to the bound states of the QW, or by release back into the SCH reservoir. Carriers captured into the bound states may return to the gateway states, or preferably, contribute to the stimulated emission of photons [43]. In [40], 2-level and 3-level rate equations models are compared and it is concluded that the impact of the carrier virtual states can be significant towards the understanding of the dynamics of QW lasers. In Figure 2.8 the lifetimes associated with virtual states (τcap and τesc ) have a quantum mechanical origin.  Capture Time:  The carriers in the continuum states need to be captured  to the QW bound states in order to contribute to the laser operation. This requires that the carrier in some initial state in the SCH region emits a longitudinal optical (LO) phonon and ends up in the some final state within the QW as dictated by energy and momentum conservation conditions [13]. Subpicosecond time-resolved measurements of the barrier luminescence decay in the GaAs/AlGaAs system have determined the quantum carrier capture and escape lifetimes [44].  Escape Time:  The dominant processes of transferring carriers into and  out of the III-V QW are via LO phonon emission and absorption [45]. Another mechanism in the escape of the carrier from the QW to the SCH region is the thermionic emission. Assuming that the carriers in the barriers have bulk-like properties and obey Boltzmann statistics the thermionic emission  39  2.4. Carrier Capture and Escape lifetime from a quantum well can be calculated as [46]:  τe =  2πm∗ d2 kB T  exp  EB kB T  ,  (2.38)  where d is the QW width, EB is the energy barrier presented to carriers in the QW, τe is the escape lifetime of the carriers from the QW, m∗ is the density of states (DOS) effective mass of the carriers, kB is the Boltzmann constant and T is the temperature in Kelvin. From equation (2.38), it is clear that the carrier escape rate is a function of the barrier height and temperature. Under ordinary conditions, hole diffusion is much slower than electron diffusion in the SCH layer. This may imply that the bandwidth limitation is due to the hole transportation. However, the quantum capture time of the holes is much shorter than the electrons [41]. For this reason we will consider only the effects of the electrons on the bandwidth. Further studies, e.g. in [47], show that capture time is constant with respect to effective barrier height (Vb in Figure 2.5), and carrier density of confined states in the QW (for the laser operation this corresponds to the threshold carrier density). This is very important in our modeling, as in a modulated laser the carrier density of the QW changes.  2.4.2  Modeling Using Quantum Capture and Escape Lifetimes  In the above approaches, the spatial distribution of the carriers in the SCH region is ignored. However, to analyze the behavior of the transistor it is 40  δn  2.4. Carrier Capture and Escape  I  N V.S  E  δn(x)  τcap  τesc  Doped Cladding  SCH X=0  Doped Cladding  NQW  QW  X = W/2  SCH  X=W  Figure 2.9: Semi-classical model for the carrier dynamics. Above part shows the carrier spatial distribution, δn(x), due to the current I. In the position of QW δn(x) = NV.S. , virtual states concentration. Below part shows the conduction band of the SCH and QW and the carriers in QW 2D states. These carriers are connected to the virtual sates carriers by τesc and τcap . W is the width of SCH region including the QW thickness. necessary to discover the spatial distribution of the minority carriers in the base [22]. This can be easily done by usage of the virtual state as a connection between the continuum states and bound states and introducing the quantum capture and escape lifetimes [48]. This model shows that in spite of their small values, these lifetimes have significant effect on the dynamics of the laser. Figure 2.9 shows the anatomy of this semi-classical approach. The spatial distribution of the carriers in the SCH region is found by  41  2.4. Carrier Capture and Escape solving the charge-density continuity equation for the excess electrons [22]: ∂δn(x, t) 1 ∂Jn (x, t) δn(x, t) , = − ∂t q ∂x τB  (2.39)  where δn(x, t) is the excess injected carrier concentration in the SCH region, Jn (x, t) is the current density due to carriers, τB is the carrier recombination lifetime (radiative and non-radiative) in the SCH region. In equation (2.39), it should be emphasized that both δn and Jn are functions of position (x) and time (t); dependance on x and t will not be written in forthcoming equations. We assume that diffusion is the dominant mechanism (neglecting the drift) for transport across the SCH region. The current term in equation (2.39) can be written as:  Jn,diffusion = qDn  ∂δn , ∂x  (2.40)  in equation (2.40), Dn is the diffusion coefficient in the SCH region. Substituting equation (2.40) into equation (2.39) we obtain the diffusion equation for excess carriers: ∂ 2 δn δn ∂δn − . = Dn ∂t ∂x2 τB  (2.41)  The injected current can be found from the slope of the carrier distribution [22]: J = qDn  ∂δn . ∂x x = 0  (2.42)  The energy barrier between the SCH region and the doped cladding causes  42  2.4. Carrier Capture and Escape the current to be zero at the right side of SCH region (ignoring the leakage current due to carrier jumping above the barrier, i.e., current efficiency of 100%): ∂δn . =0 ∂x x=W  (2.43)  The virtual states’ carrier concentrations and current density can be described by the following equations [48]:  δn (  JV.S. = qDn  W − ) 2  ∂δn ∂x  = δn (  W x = ( )− 2  W + ) 2  − qDn  ∂δn ∂x  = NV.S. ,  W x = ( )+ 2  (2.44)  ,  (2.45)  where, NV.S. is the carrier concentration in the virtual states. Equation (2.44) states that the concentration profile δn(x) is a continuous function and equation (2.45) states that the current flowing to virtual states (JV.S. ) is the difference of the continuum states currents at x = W/2. The laser is described by the conventional laser rate equations [11], in terms of the current entering the QW region: dS = dt  ΓG −  1 τP  S + Rsp ,  dNQW JQW NQW − GS , = − dt qd τS G=  G0 (NQW − Ns ) , 1+ S  (2.46)  (2.47) (2.48)  in equations (2.46)-(2.48), S is the photon concentration, NQW is the QW  43  2.4. Carrier Capture and Escape carrier density, Γ is the optical confinement factor, G is the optical gain, τP is the photon lifetime, Rsp is the spontaneous emission that is coupled to the lasing mode and is found from equation (2.7), JQW is the current density entering the 2D QW energy states from the virtual states, d is the QW width, τS is the spontaneous emission lifetime, is the gain compression factor due to spectral hole burning, spatial hole burning, and carrier heating. Ns is the fitting parameter of the optical gain function. The carriers entering the virtual states have three possibilities: falling into the QW states, or diffusing to the SCH region or undergoing recombination (radiative and non-radiative). The rate equation describing the virtual states concentration is: dNV.S. JV.S. JQW NV.S. . = − − dt qd qd τS  (2.49)  Equation (2.49) describes the charge conservation for the virtual states in the QW region and is proven in Appendix B. The virtual state carriers are linked to the QW 2D carriers by: JQW NV.S. NQW − , = qd τcap τesc  (2.50)  where τcap is the quantum capture lifetime for the carriers falling from the virtual states to the QW 2D states, and τesc is the escape lifetime from the QW 2D states to the virtual states. Equations (2.40) to (2.50) are a complete set describing the laser performance. We solve these equation in different regimes, e.g., small-signal  44  2.4. Carrier Capture and Escape and large-signal for different devices, e.g., SCH laser and TL using different method, e.g., analytical and numerical. These are the main the contributions of this study and are discussed in the rest of the thesis. The solving procedure starts with replacing equation (2.41) with its finite difference approximation. We use the following notation:  p Cm = C(m, p) = δn(mΔx, pΔt) ,  (2.51)  where m and p are used as indices to x and t, respectively and Δx and Δt are the spatial and temporal grid size, respectively. Using the finite difference approximations for the derivatives: p p − Cm−1 Cm , Δx  (2.52)  p p p Cm+1 − 2Cm − Cm−1 , (Δx)2  (2.53)  p p−1 − Cm Cm , Δt  (2.54)  ∂δn ∂x ∂ 2 δn ∂x2  ∂δn ∂t equation (2.41) is converted to:  −  Dn Δt p Dn Δt Δt C + 1+2 + Δx2 m−1 Δx2 τB  p Cm −  Dn Δt p p−1 C = Cm . (2.55) Δx2 m+1  In equations (2.53), and (2.54), we are using backward difference for the time derivative term and second-order central difference for the space derivative. This is an implicit method for the 1D continuity equation. The method is always numerically stable and convergent but it is more numerically intensive  45  2.4. Carrier Capture and Escape than the explicit method [32], where forward difference is used for the time derivative term. The numeric error due to finite difference for time derivative is proportional to the temporal grid size, i.e., O(Δt) and for second order spatial derivative it is proportional to O(Δx2 ). The boundary conditions are applied to equation (2.55) and knowing the initial values of the parameters and injected current a matrix structure of equation (2.55) is formed:  AC(p) = B(p−1) ,  (2.56)  where A and B are known matrices. A is a constant coefficient matrix which is computed once for a specific device. Defining κ, η, μ, and ξ as:  κ=  η =1+2 μ= ξ =1+2  Δt , τB  (2.57)  Dn Δt Δt + , Δx2 τB  (2.58)  Dn Δt , d Δx  (2.59)  Δt Dn Δt Δt + . + d Δx τcap τS  (2.60)  46  2.4. Carrier Capture and Escape A will have the following format: ⎛  Am,n  ⎞  0 ··· 0 0 ⎜ 1 −1 0 ⎜ ⎜−κ η −κ 0 ··· 0 0 ⎜ ⎜ ⎜ ⎜ 0 −κ η −κ · · · 0 0 ⎜ ⎜ . .. .. .. .. .. .. ⎜ .. . . . . . . ⎜ =⎜ ⎜ 0 · · · −μ ξ −μ · · · ⎜ 0 ⎜ .. .. .. .. .. ⎜ .. .. ⎜ . . . . . . . ⎜ ⎜ ⎜ 0 0 0 0 · · · −κ η ⎜ ⎝ 0 0 0 0 ···0 1  0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ . ⎟ 0 ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ −κ⎟ ⎟ ⎠ −1  μ and ξ are the parameters relating the continuum states’ carriers to QW carries. Values of B depend on the parameters at t = (p − 1)Δt and the pumping current and it has the following form: ⎞  ⎛  Bm  ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜C p−1 + ⎜ v.s ⎜ ⎜ ⎜ ⎜ ⎝  Δx J Dn e  ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ C3p−1 ⎟ ⎟ .. ⎟ . . ⎟ ⎟ Δt Nqw (p − 1)⎟ ⎟ τesc ⎟ ⎟ .. ⎟ . ⎟ ⎠ p−1 CM C2p−1  Solving equation (2.56) gives C. Having the carrier distribution in the SCH region, the other parameters can easily be obtained. Figure 2.10 shows the flowchart for solving the equations. The solving procedure starts with the  47  2.4. Carrier Capture and Escape initialization of the matrices and depending on DC or AC simulation the current vector is updated. The variables are calculated in the simulation time period and in the final stage all the parameters are found. The simuStart Get the laser and transistor parameters  Initialize the matrices (A, B, C)  DC or AC  AC  Add RF signal to DC bias current  DC Find S and Nqw from rate equations using finite difference method  Next f  Update B and find C, C=A\B  Next t  No  No t=t_max Yes DC or AC  AC  All frequencies are swept?  DC  Find the transistor currents and optical power  Yes  End  Figure 2.10: Flowchart of the simulation. The core of the simulation is based on the finding the carrier concentration, matrix C, in every time step for each value of the current or frequency. lation time is characterized by tmax in the flowchart. tmax values depend on the type of the simulation, i.e., it should be long enough to let the transient 48  2.4. Carrier Capture and Escape portion go away. In the DC analysis, tmax is 2 - 5 ns, in AC analysis tmax ∼ 50 ns, and in digital modulation tmax = nT where n is the number of the bits and T is the bit’s period. Table 2.2 shows the values of the additional parameters used in the simulation. We consider a symmetrical structure where the QW is placed in the middle of the SCH region. In practical situations asymmetrical SCH regions (different doping and different length) are used to reduce the carrier transfer effect [49]. Parameter Diffusion coefficient Quantum capture lifetime Quantum escape lifetime SCH region recombination lifetime SCH region thickness QW thickness QW position  Unit cm2 s−1 ps ps ns nm nm nm  Symbol Dn τcap τesc τB dsch dQW x0  Value 80 1 50 1 88 12 50  Table 2.2: Typical simulation laser parameters. The values are chosen for typical 980 nm VCSELs. Figure 2.11 shows the carrier distribution in the SCH layer. The energy barrier at X = W = 100 nm blocks the carriers from further diffusion beyond the SCH region and the distribution has a zero slope. As the injection current increases the magnitude of the distribution increases and the slope of the distribution on the left side increases. An interesting feature of this figure is the effect of the stimulated emission on the carrier distribution. Above threshold, the carrier distribution does not grow as fast as below threshold. Figure 2.12 shows the small-signal modulation response of the same laser 49  2.4. Carrier Capture and Escape 2.2  1.8 1.6  δn (10  17  −3  cm )  2  1.4  I = 0.5 mA I = 1 mA I = 2 mA I = 3 mA  1.2 1 0.8 0  20  40 60 x (nm)  80  100  Figure 2.11: Carrier distribution at the different bias currents. The compression of the carrier distribution profile above laser threshold is evident by the decreased spacing of the curves. The simulated device has a threshold of Ith = 0.6 mA. that was simulated with 1- and 2- level rate equations at different bias currents. Further damping in the response is evident and it is explained by the effect of the carrier transport. Here we investigate the effect of the carrier transport on the dynamics of the laser. Figure 2.13 shows the small-signal bandwidth variation for two different quantum capture lifetimes and the same escape lifetime as a function of bias current. Increase in the bandwidth with decreasing the capture time is evident. In general the capture lifetime of the holes is smaller than the electrons [41] so the limitation on the small-signal modulation bandwidth mainly comes from the electrons. From Figure 2.13 the bandwidth has a maximum value at a specific bias current and beyond that the bandwidth saturates at a high bias current, this phenomenon comes from the damping,  50  2.4. Carrier Capture and Escape  10  Transfer Function (dB)  0 −10 −20 I = 1 mA I = 2 mA I = 3 mA I = 10 mA  −30 −40 −50 −60 0.1  1  10  30  f (GHz)  Figure 2.12: Simulation results for the small-signal modulation response of a 980 nm VCSEL at different bias points by using the semi-classical model. The amplitude of the RF current is set to im = 10 μA.  16  Bandwidth (GHz)  14 12 τcap = 1 ps  10  τ  cap  8  = 0.6 ps  6 4 2 0  5  10  15  I (mA)  Figure 2.13: Small-signal modulation bandwidth variation of a 980 nm VCSEL versus DC bias current.  51  2.5. VCSEL Parasitic Modeling gain compression, carrier transport and diffusion. In the section 2.6, we will see that this model leads to a good agreement with the experimental data.  2.5  VCSEL Parasitic Modeling  In VCSELs the RF signal passes through a number of resistors and capacitors before reaching the active region. The important resistors are: the nand p- DBRs resistance, contact-wafer resistance, and sheet resistance between the bottom DBR to the contact [49]. The main capacitors are oxide layer capacitance and junction capacitance. The effect of the parasitics can be modeled approximately by a low-pass single pole transfer function [49]. This is due to the capacitive and resistive elements which can be approximated by the equivalent RC circuit. The justification for the use of a single pole transfer function for the parasitics comes from the fact that it can accurately represent the small-signal modulation response over a large range of currents [50]. The overall transfer function can be written as:  H(ω) =  isch p(ω) p(ω) = , is is isch  (2.61)  where is is the current from the electrical source, isch is the current entering the SCH region. isch /is is the parasitic term and the next term is the transfer function from the model. By using semi-classical modeling, writing the transfer function of smallsignal modulation of the laser analytically in a closed form similar to the equations (2.23) and (2.35) requires much effort. We will do this in the  52  2.6. Model Verification next chapter. Meanwhile we can approximate the small-signal modulation response by a third order transfer function similar to equation (2.35). Considering the parasitic effect as a first order transfer function, the normalized transfer function can be written as:  H(ω) =  1 1 ω · ω · ω 2 γ ω , 1+j 1+j 1−( ) +j · ωpar ωa ωr 2πωr ωr 1  (2.62)  where ωpar is the parasitic pole, ωa is the pole due to the carrier dynamics, gain compression, and γ and ωr are the damping and relaxation oscillation frequency, respectively.  2.6  Model Verification  In this section we show that the modeling based on the rate equations can match the real experimental results. These results are obtained for 850 nm VCSELs.  2.6.1  Sample Preparation  Conventional VCSELs operating at 850 nm and 980 nm have been successfully fabricated by Dr. Mark Greenberg, a Research Associate, and their DC and AC characteristics have been measured. The necessary layer structures for the laser operation have been designed. The wafers were ordered from different companies and research centers. Most of the processing steps including metallization, etching, oxidation, planarization are done in the UBC AMPEL nano-fabrication facility. Figure 2.14 shows a high speed 850  53  2.6. Model Verification nm VCSEL working in different regimes of biasing. The device is not oxidized so the threshold is high. This figure shows the basic method used to find out if the device is working. It uses an infrared camera so that the user can see the light, as in Figure 2.14 (b) and (c). Above threshold the user can see the interference pattern due to reflection of the laser light from the microscope lenses, as in Figure 2.14 (d). The samples are oxidized at  (a)  (b)  (c)  (d)  Figure 2.14: High speed 850 nm VCSEL in different biasing regimes. a) No biasing b) I = 1 mA, spontaneous emission c) I = 10 mA, just below threshold and d) I = 15 mA, above threshold. different temperature and for different amounts of times. Oxidation layers are used in the VCSEL design to reduce the threshold current of the laser by confining the current to a smaller area and hence increasing the current 54  2.6. Model Verification density. Aluminum-oxide (AlOx) is used to accomplish this goal. AlOx is an electrical insulator which is highly selective on Al composition, and has low refractive index. AlOx is formed from high Al composition (x > 0.9) of Alx Ga1−x As [51]. Oxidation is done in a furnace at a temperature of >400 ◦ C.  The oxidation is a one-run process.  2.6.2  DC Measurements  The DC setup consists of DC probes for the biasing of the laser and a Keithley 2602 unit which is used as a precise current source to bias the laser. A Newport 2830C power-meter and a Newport 818IR photo-detector are used for the optical power measurement. A wide aperture lens is used to collect almost all of the optical power to increase the power-reading accuracy. An Ando AQ6317B optical spectrum analyzer is used for the spectrum measurements. Figure 2.15 (a) shows the voltage and output power against the bias current (L-I-V). The laser diode turn-on voltage, a good measure of the excess voltage drop from the hetero-barriers of the DBRs, is 1.4 V. This low threshold voltage is the consequence of the low threshold current. The device has a slope efficiency of 0.33 W/A corresponding to a differential efficiency of 29% and current threshold of ∼ 1 mA. Figure 2.15 (b), shows the normalized optical spectrum of the 850 nm VCSELs. It is a single mode laser with side mode suppression ratio (SMSR) of > 40 dB.  55  2.6. Model Verification  6  0  3  2  2  1  Amplitude (dB)  4  Voltage (V)  Optical Power (mW)  −10 −20 −30 −40 −50 −60 0 0  5  10 Current (mA)  15  (a)  0 20  800  820 840 860 Wavelength (nm)  880  (b)  Figure 2.15: DC measurement of a 850 nm VCSEL. a) L-I-V curves of the 850 nm VCSEL at T = 25 o C. b) Measured optical spectrum of 850 nm VCSEL. The spectrum has been normalized.  2.6.3  AC Characteristics  Figure 2.16 shows the AC setup used for the AC measurements. It consists of a bias-T to add the RF modulation signal to the DC current. The VCSEL was probed on the wafer using a 40 GHz probe and a lensed fiber was used to collect the optical power from the VCSEL. The small-signal modulation response (S21 ) was characterized using an Agilent E8361A 67 GHz network analyzer using a resolution bandwidth of 10-1000 Hz. The modulated VCSEL output was directly detected using a high speed Discovery Semiconductor Inc. photo-detector (bandwidth of 10 GHz) with a known S21 . A through calibration of the network analyzer was performed to calibrate the frequency response of the cables. The frequency response of the photo-detector, measured independently, was subtracted from the S21 data. Thus, the frequency responses data reported here are the actual response of the VCSEL including device parasitics. The RF power is set to a small 56  2.6. Model Verification  DC  Bias-T  Probe  PD  Optical Coupler  Optical Fiber  OSA  VNA  VCSEL  RF signal  Figure 2.16: AC measurement experimental setup. Laser is biased by a high speed probe. AC and DC currents are applied by using a BiasT. The light of the laser is collected by a fiber and a high speed photo-detector converts the modulated light to the electrical current. RF response is measured by a network analyzer. value (-20 to -15 dBm) to make sure that small-signal modulation regime is achieved. Figure 2.17 shows the measured RF response for different bias currents. The device under test had a threshold of 0.9 mA. The low DC gain is explained by the large coupling loss and low responsivity of the detector at the operation wavelength. The frequency response of the detector has been de-embedded from the final result. As it is clear from the figure, increasing the bias current will first increase the resonance peak and then decrease it as the model predicts. Further insight can be obtained by the curve-fitting of the measurement results to the equation (2.62). First the parasitic pole is found. One method of determining the device parasitics of the VCSEL is by de-embedding the parasitics term from the modulation response [52]. In this method, the 57  2.6. Model Verification −30 −40 RF Response (dB)  −50 −60 −70 −80 I = 1.1 mA I = 2.5 mA I = 4.5 mA  −90 −100 −110 0  2  4  6  8  10  f (GHz)  Figure 2.17: Measured small-signal modulation responses at different current injection levels. The low-frequency fluctuations are mainly from the detector used in measurements. modulation response is measured at two bias conditions, assuming that the device parasitics are independent of the bias current, by dividing the two responses we can get rid of the first term in equation (2.62) and the result will be: 1  |H2 (f )|2 /|H1 (f )|2 = 1+  f fa1  2  1  · 1−  1 1+  f fa2  2  f fr1  2  2  γ1 f · 2πfr1 fr1  +  2  1  · 1−  f fr2  2  2  +  γ2 f · 2πfr2 fr2  2  ,  (2.63) where fa1 is the pole due to carrier dynamics, fr1 is the relaxation oscillation frequency, and γ1 is the damping at first bias current. fa2 is the pole due  58  2.6. Model Verification to carrier dynamics, fr2 is the relaxation oscillation frequency, and γ2 is the damping at second bias current. We apply this method to the measurement results. Figure 2.18(a) shows the measured frequency responses (Hm,1 and Hm,2 ) of the small-signal modulation at two different bias currents. These responses are in dB scale and we find the difference response by subtracting the modulation response, i.e, Hm,2 − Hm,1 . This is shown in the Figure 2.18(b). We fit the equation (2.63) into the difference response and find the values of the parameters of the model. Table 2.3 summarizes the results of the curve-fitting to the measurements of Figure 2.18(a). It is worth noting that the accuracy of the pole due to carrier transport, i.e., fa1 and fa2 , is not high. It is partly due to the finite dynamic range of the measurement system. To accurately find those poles we need to measure the modulation responses to the higher frequencies (> 50 GHz). Parameter fr fa γ  I = 4 mA 5.6 13.3 13  I = 7.5 mA 8 14 24  Unit GHz GHz GHz  Table 2.3: Extracted values for the laser modulation transfer function. These values are for the model parameters in the Equation (2.62). Using the Table 2.3 we construct the model based transfer functions Ht,1 : 1  Ht,1 = 1+  f fa1  2  1  · 1−  f fr1  2  2  +  f γ1 · 2πfr1 fr1  2  . (2.64)  The parasitics response can be obtained by comparing the measurement 59  −40  10  −50  5  Measurement Model  Hm,1 −60  Hm,2  0  1  H −H (dB)  −70  −5  2  RF response (dB)  2.6. Model Verification  −80 −90 −100 0  −10 I = 4 mA I = 7.5 mA  5  10 f (GHz)  −15  15  20  −20 0  5  10 f (GHz)  (a)  15  20  (b)  Figure 2.18: Parasitic estimation method. a) Two measured frequency response curves at different bias current. b) Difference between the responses. The measurement curve is obtained by the subtracting the measurement responses and the model curve is obtained by curve-fitting equation 2.63 to the measurement curve. results of Figure 2.18(a) and model based transfer function:  Hpar = Hm,1 − Ht,1 .  (2.65)  We have assumed that the parasitics response is independent of the bias point. This can be checked by calculating the parasitics response at different bias point and compare the results. Figure 2.19(a) shows the parasitics of the modulation responses in Figure 2.18(a) and the analytical result: 1  |Hpar (f )|2 = 1+  f  2  ,  (2.66)  fpar  where fpar is the parasitic response 3-dB bandwidth. Ignoring the low frequency oscillations which are partly from calibration and partly from  60  2.6. Model Verification high speed photo-detector response, there is a good agreement between two curves.  0 −2 −4 −6 −8 −10 0  −45  Measurement Model Modulation Response (dB)  Parasitic Response (dB)  2  5  10  15  −50 −55 −60 −65 −70 −75 0  I = 7.5 mA, Measurement I = 4 mA, Measurment I = 4 mA, Model I = 7.5 mA, Model 5  10  f (GHz)  f (GHz)  (a)  (b)  15  Figure 2.19: Parasitic estimation method. a) Parasitic frequency response. The measurement curve is obtained by using equation (2.65) and the model curve is obtained by using equation (2.66) and curve-fitting. From the model curve fpar = 8 GHz is obtained. b) Parasitic-free modulation responses. The measurement curves are obtained by subtracting the measured parasitic response from the curves of Figure 2.18(a) and the model curves are obtained by using the equation (2.64) and values of Table 2.3.  Finally, subtracting the parasitic response from the measured data, the parasitic-free modulation response is obtained and it is shown in Figure 2.19(b).  61  Chapter 3  Transistor Laser ∗ 3.1  Bipolar Junction Transistor  In this section we review the fundamental principles of the bipolar junction transistors (BJT). This introduction will help in understanding the TL. A BJT is a three terminal device made of two back-to-back simple p-n junctions. As shown in the Figure 3.1 (a) the forward-biased n-p junction (emitter junction) injects electrons into the center p region (base). These minority carriers (electrons) make the reverse current through the p-n junction (collector junction). A few observations can be made from this figure. First, it is important that electrons can diffuse to the depletion layer of the reversed-biased junction before they recombine in the base layer. For this reason the base layer should be narrow. In general Wb  Ln , where Wb is  the length of the base region and Ln is diffusion length of the electrons in the base: L2n = Dn τn ,  (3.1)  ∗  A version of this chapter has been published: B. Faraji, W. Shi, D. L. Pulfrey, and L. Chrostowski, “Analytical modeling of the transistor laser,” IEEE Journal of Selected Topics in Quantum Electronics, vol. 15, pp. 594-603, 2009.  62  3.1. Bipolar Junction Transistor VBE  VCB  IB  IB  Base IE n+  IC p  Emitter  n  Collector Wb  Electron collected  Electron injected  (a)  4 IE  3  6 IC  5  2  1 n+  p  n  (b)  Figure 3.1: An n-p-n transistor. a) Schematic of an n-p-n transistor. Forward-biased emitter junction injects the electrons to the narrow base, where electrons diffuse to reversed-biased collector junction and contribute to the collector current. b) Major current components in an n-p-n transistor: (1) injected electrons reaching the reversed-biased collector junction, (2) electron lost in the base due to recombination with the holes in the base, (3) holes supplied by the base terminal, (4) holes that are injected to the emitter through the forward-biased junction, (5) thermally generated electrons in the base reaching to the reversed-biased junction, and (6) thermally generated holes in the collector reaching to the reversed-biased junction. Components (5) and (6) make the saturation current at collector junction. where Dn is the diffusion coefficient of the electrons in the base and τn is the recombination lifetime. Second, the emitter current, IE , should be composed mainly of the electrons injected to the base. This requirement can be achieved by increasing the doping of the emitter region which results in a n+ -p junction. Base current results from different components 1) recombination of the injected electrons from the emitter with the holes in the base 2) injection of the holes to the emitter 3) injection of the thermally generated holes in the collector to base. The dominant process is usually the recombination and we will ignore the other two mechanisms in the modeling.  63  3.1. Bipolar Junction Transistor Figure 3.1 (b) summarizes the major electron and hole current components. The emitter injection efficiency, γ, is defined as the ratio of the desired forward emitter current to the overall emitter current:  γ=  iEn , iEn + iEp  (3.2)  where iEn is summation of currents of the arrows (1) and (2) in the Figure 3.1 (b) and back injection current iEp is the current associated with arrow (4) in the Figure 3.1. Ignoring the saturation current at the collector junction, the base transport factor, B, is the defined as the ratio of the collector current to the desired emitter current:  B=  iC , iEn  (3.3)  where iC is the collector terminal current. To have a good n-p-n transistor, the emitter current should be made up mainly from the electrons injected to the base and these electrons should reach to collector before significant recombination, in the other words γ and B should be very close to one. Using the above parameters, we can relate the collector to the base current in the following manner:  BiEn iC = iB iEp + (1 − B)iEp =  iEn iEn + iEp = iEn 1−B iEn + iEp B  (3.4)  Bγ ≡β. 1 − Bγ  64  3.2. Transistor Laser Modeling In equation (3.4), β is the base to collector amplification factor. For welldesigned transistors β is large (100 - 200). A p-n-p BJT is treated in the same way that an n-p-n BJT was studied with the exception that the roles of the electrons and the holes are switched.  3.2  Transistor Laser Modeling  In the TL a light source (e.g., a QW) is embedded in the base of a heterojunction bipolar transistor (HBT) which acts as an optical collector. One interesting feature of the TL is the potential for an enhanced small-signal modulation bandwidth due to the reduced carrier lifetime in the base region. The reduced carrier lifetime is due to the reverse biased base-collector junction which introduces a gradient in the carrier concentration (as shown in Figure 3.2). The physical parameter associated with this slope is the base transit time (τt ) which is the average time an electron spends in transit across the base. For the modeling of the TL we use the concepts developed in Chapter 2. The concept of virtual states [43] is necessary to obtain a useful amalgamation of the laser rate equations and the diffusion equation. We consider an N-p-n HBT. We use the convention that the capital letter shows the larger energy band gap, in this case the material used in emitter has larger energy gap than the base material. The choice of HBT comes from its very high emitter injection efficiency. As the back injection current experiences large energy barrier because of the hetero-structure in the emitter-base junction, γ can be very close to 1.  65  3.2. Transistor Laser Modeling  Figure 3.2: Schematic of carrier diffusion and quantum capture in the QW, and the conduction band energy diagram of the base region. The emitter is at the left side of the base (x < −WB /2) and the collector is at the right side (x > WB /2). We assume that the transistor is operating in its normal, active mode, i.e., the base-emitter junction is forward-biased and the base-collector junction is reversed-biased. Figure 3.2 shows the conduction energy band of the base and the excess minority carrier distribution, δn(x), in the base region. We assume that there is only one QW which is located in the middle of the base region and acts as a source for the laser emission. The generalization for more that one QW will be discussed in Section 3.7. We assume that the base is neutral and the excess majority carriers are provided through the base contact to maintain the charge neutrality. However, the collector current is mainly due to excess minority carriers. This is the reason of tracking the minority carriers in the base of the TL. The carriers injected from the emitter diffuse across the base and reach to the QW. These unbounded carriers may undergo quantum capture to the bound states in the QW with a lifetime of τcap , or diffuse across to the  66  3.2. Transistor Laser Modeling collector where they are swept out by the reverse-biased base-collector junction. The carriers may escape the QW with a lifetime τesc . The unbounded carriers at x = 0 are located at the virtual bound states. These states are localized at the QW, but occupy energies higher than the conduction energy band edge of the barrier material and aid in the conversion of carriers from the 3D states (nV.S. ) above the well to the 2D states within the QW (nQW ) and vice versa [43]. The QW adds another recombination region in the base which has a variable recombination rate. Below laser threshold, carrier removal from the QW is dominated by the spontaneous emission which is characterized by the lifetime of τS , while above the laser threshold the carriers are removed from the QW through the stimulated emission. From simple rate equation the effective recombination lifetime of the QW carriers can be written as: 1 τr,eff  =  1 GS + . τS Nth  (3.5)  Equation (3.5) states that, above the threshold, carriers are removed faster as the output power of the laser increases. The base current consists of two components: 1) the regular base region recombination current which is characterized by a base region recombination lifetime (τB ) and a diffusion coefficient(Dn ), 2) the current needed to drive the laser. We derive expressions for both currents.  67  3.2. Transistor Laser Modeling  3.2.1  General Consideration  The key point in the analysis of the bipolar transistors is to find the excess minority carriers distribution in the base region which are electrons in the case of an N-p-n HBT. For this purpose we solve the continuity equation for the injected electrons to the base region [22]: ∂δn(x, t) 1 ∂Jn (x, t) δn(x, t) , = − ∂t q ∂x τB  (3.6)  where δn(x, t) is the excess carrier (electrons) concentration in the base, Jn (x, t) is the current density due to excess carriers (electrons), τB is the carrier recombination lifetime in the base region, and q is the electronic charge. In equation (3.6), δn and Jn are functions of position (x) and time (t); we will not write the x and t dependency in forth-coming equations. We assume that diffusion is the dominant mechanism (negligible drift) for transport across the base. This assumption is valid in most situations where the base is heavily doped, and is discussed further in Section 3.7. We can replace the current term in equation (3.6) by the expression for the diffusion current, i.e.: Jn, diffusion = qDn  ∂δn , ∂x  (3.7)  in equation (3.7), Dn is the diffusion coefficient of the electrons in the base. Substituting equation (3.7) into equation (3.6) we obtain the diffusion equation for excess carriers: ∂ 2 δn δn ∂δn − . = Dn ∂t ∂x2 τB  (3.8)  68  3.2. Transistor Laser Modeling The emitter and collector current densities are found by ∗ :  jE = qDn  ∂δn ∂x  jC = qDn  ∂δn ∂x  ,  WB x=− 2  WB x= 2  ,  (3.9)  (3.10)  where the WB is the width of the base. If the collector junction is strongly reversed biased (VCB  0, Figure 3.1), the excess carrier concentration on  the collector side of the base is found from [22]:  δn(  WB ) = np (eVCB /kT − 1) 2  (3.11)  = −np , where np is the base equilibrium hole concentration. In equation (3.11), if we assume that the equilibrium hole concentration is negligible compared with the injected concentration δn(−WB /2), equation 3.11 is reduced to:  δn(  WB )=0. 2  (3.12)  The equations for the carrier concentration (nV.S. ) and the current density (jV.S. ) of the virtual states located at x = 0 are: δn(0− ) = δn(0+ ) = nV.S. ,  (3.13)  ∗  Small letters with capital subscripts are used for total quantities (DC+AC); capital letters with capital subscripts are used for DC quantities; and small letters with small subscripts are used for the AC terms.  69  3.2. Transistor Laser Modeling  jV.S. = qDn  ∂δn(0− ) ∂δn(0+ ) − qDn . ∂x ∂x  (3.14)  Equation (3.13) states that the concentration profile δn(x) is continuous function and equation (3.14) states that the current flowing to virtual states (jV.S. ) is the difference of the continuum states currents at x = 0. The current relation of the transistor is:  jE = jC + jB ,  (3.15)  where jB is the base current density. Equation 3.15 is proven in Appendix A. The laser is described by the conventional laser rate equations [11], in terms of the current entering the QW region: dS = dt  ΓG (nQW , S) −  1 τP  S + Rsp ,  dnQW jQW nQW − G (nQW , S) S , = − dt qd τS G (nQW , S) =  G0 (nQW − Ns ) . 1+ S  (3.16)  (3.17) (3.18)  In equations (3.16) - (3.18), S is the photon concentration, nQW is the QW carrier density, Γ is the optical confinement factor, G(nQW , S) is the optical gain, τP is the photon lifetime, Rsp is the spontaneous emission that is coupled to the lasing mode, jQW is the current density entering the QW 2D energy states from the virtual states carriers, d is the QW width, τS is the spontaneous emission lifetime,  is the gain compression factor, and Ns is  70  3.2. Transistor Laser Modeling the fitting parameter of the optical gain function [11]. The carriers entering the virtual states have two possibilities: falling in the QW states, or diffusing to the collector. The rate equation describing the virtual states concentration is: dnV.S. jV.S. jQW nV.S. . = − − dt qd qd τS  (3.19)  Equation (3.19) describes the charge conservation for the virtual states in the QW region and is proven in Appendix B. The virtual state carriers are linked to the QW carriers by: jQW nV.S. nQW − , = qd τcap τesc  (3.20)  where τcap is the capture lifetime for the carriers falling from the virtual states to the QW 2D states, and τesc is the escape lifetime from the QW 2D states to the virtual states. The numerical values used in the model are summarized in Table 3.1. The values are chosen to describe a transistor vertical cavity surface emitting laser (TVCSEL), e.g., [53]. The values of the τcap and τesc used in the simulations are for SCH laser. However, their usage in the TL simulations is justified with the argument of Section 2.4 as the value of τcap does not change with carrier density variation and the background carrier density due to the diffusion of the holes from heavily doped base to the QW, i.e., doping modulation effect [54]), can be ignored in comparison with the carrier densities above threshold.  71  3.3. DC Analysis Parameter Mesa size Spontaneous emission lifetime Photon lifetime Quantum capture lifetime Quantum escape lifetime Base recombination lifetime Diffusion coefficient Linewidth enhancement factor Group velocity Mirror loss Internal loss Active area thickness Optical confinement factor Fitting parameter Temporal growth coefficient spontaneous emission coupling coefficient Active region refractive index Gain compression factor  Symbol A τS τP τcap τesc τB Dn α vg αm αi d Γ Ns G0 β ng  Value 9π 500 3 1 20 200 26 3 5.13 40 25 12 0.05 1.2 × 1018 3 × 10−6 10−4 5.8 1.5 × 10−17  Unit μm2 ps ps ps ps ps cm2 /s cm/ns cm−1 cm−1 nm cm−3 cm3 s−1 cm3  Table 3.1: Values of the parameters used in the simulations. The values are chosen for a 980 nm TVCSEL.  3.3  DC Analysis  DC analysis is done by setting all the time derivatives to zero in the equations described in the previous section. We ignore the coupling of the spontaneous emission to the lasing mode. This assumption does not affect the generality of the solutions but makes the mathematical derivation simpler. Using equation (3.8) and the above mentioned boundary conditions the  72  3.3. DC Analysis carrier concentrations can be found:  δN1 =  N V.S. eWB /2LD −  LD qDn JE x/LD  e  WB 2LD  2 cosh  NV.S. e−WB /2LD +  +  LD qDn JE −x/LD  e  WB 2LD  2 cosh  ,  (3.21)  δN2 =  −NV.S. e−WB /2LD 2 sinh  WB 2LD  ex/LD +  NV.S. eWB /2LD 2 sinh  WB 2LD  e−x/LD ,  (3.22)  where δN1 and δN2 are the DC carrier concentrations in the regions before the QW and after the QW, respectively; LD is the diffusion length defined as L2D = Dn τB . We can find transistor currents, JE , JC and JB , in terms of JV.S. and NV.S. : ⎛ JE = NV.S.  qDn ⎝ sinh LD  WB 2LD  cosh2  WB 2LD  + sinh  WB 2LD  ⎞ ⎠ + JV.S. cosh  WB 2LD  , (3.23)  ⎛ JC = NV.S.  JB = 2NV.S.  qDn sinh LD  cosh2  qDn ⎝ LD sinh WB 2LD  WB 2LD WB 2LD  ⎞ ⎠ ,  WB 2LD  + JV.S. cosh  recombination current  (3.24)  .  (3.25)  laser current  Equation (3.25) states that the base current has two components. The base current of a BJT is in the form of [22]:  JB = ΔnE  qDn sinh LD  WB 2LD  ,  (3.26)  73  3.3. DC Analysis where ΔnE is the change of the minority carrier density of the base at x = 0 due to injection of the carriers from emitter. Comparison of equations (3.25) and (3.26) shows that the first term in equation (3.25) is the (radiative or non-radiative) recombination of the carriers. We call this term the recombination current. The second term is the additional (radiative) recombination due to the laser operation and we call it the laser current. The recombination current is the dominant term below laser threshold (IB < IB,th ) because there is no stimulated emission in the QW and we may ignore the laser current. For currents well above threshold (IB > IB,th ), the laser current is much larger than the recombination current. The semiconductor optical gain function is approximated by its Taylor expansion about the threshold point, i.e., NQW = Nth and S = 0. For the cases considered, the term S0 is small, and we can write: G (NQW , S0 ) =  G0 (NQW − Ns ) 1 + S0  ≈ G0 (NQW − Ns ) (1 − S0 )  (3.27)  ≈ G0 (Nth − Ns ) + G0 (NQW − Nth ) − G0 (Nth − Ns ) S0 . Using equation (3.16) and neglecting spontaneous emission, it can be shown that: Gth = G0 (Nth − Ns ) =  1 , ΓτP  (3.28)  which states that at threshold, the optical gain equals the loss in the cavity. Using equations (3.27) and (3.28), we can write the DC rate equations. From  74  3.3. DC Analysis equation (3.17): 0=  JQW NQW S0 − , − qd τS ΓτP  (3.29)  and from equations (3.27) and (3.17):  0 = G0 (NQW − Nth ) −  S0 . ΓτP  (3.30)  From equations (3.29) and (3.30), S0 may be found in terms of the QW current and the threshold carrier concentration:  S0 =  1 ΓτP JQW − Nth qd τS 1+  .  (3.31)  G 0 τS  Using equations (3.19), (3.20), (3.25), and (3.31), the DC analysis is completely described. Figure 3.3(a) shows the minority carrier distribution in the base of the transistor for different base currents. Below threshold, the spontaneous emission is the dominant process in converting the QW carriers to propagating photons. This explains the similarity of the carrier distribution of the TL to that in the base of a BJT [22]. As base current increases, the stimulated emission dominates and the current to the QW (JQW ) increases to supply the large carrier removal from QW, and the carrier distribution curves bend slightly at the location of the QW. The bending of the carrier distribution above threshold in Figure 3.3(a) is not obvious due to the large current gain of the transistor near laser threshold (β ≈ 50). Figure 3.3(b) shows the carrier density distribution for  75  10 −3  cm )  5  I  =1.5 mA  B,th  3  Δ IB=0.6 mA  QW  2 4.5 mA 1 0  0.3 mA 0  20 40 60 80 Position in the base (x, nm)  (a)  8  100  I  =1.1 mA  B,th  16  4  Carrier Density (δ n, 10  Carrier Density (δ n, 1016 cm−3)  3.3. DC Analysis  Δ I =0.45 mA B  6  QW 4  3.4 mA 2  0.2 mA 0 0  50 100 150 Position in the base (x, nm)  200  (b)  Figure 3.3: a) Calculated minority carrier density distribution, plotted for uniformly spaced base current values. The compression of the carrier distribution profile after laser threshold is evident by the decreased spacing of the curves. The red curve, labeled IB,th = 1.5mA, is the carrier density distribution at threshold. The calculated threshold current is IB,th ≈ 1.5 mA. The compression can be understood by comparing the value of the emitter current IE for different values of base current IB . At IB,th , IE = 76 mA and at IB = 4.5 mA, IE = 92 mA. This compression corresponds to the decrease in βDC . This graph is for the carrier density in the continuum energy states. The 2D bound carrier density in the QW is not shown here. b) Carrier density distribution for a low current gain TL. Above threshold the kink in the profile is obvious. another TL. Design parameters are chosen so that the transistor current gain is small (β ≈ 5). Above laser threshold, the kink in the carrier density distribution profile is obvious. Figure 3.4 shows the output power and variation of the DC current gain (βDC = IC /IB ) and AC current gain (βAC = ΔIC /ΔIB ) as a function of the base current. Laser stimulated emission above threshold is the reason for the current gain decrease in Figure 3.4. Above threshold, the laser current in equation (3.25) increases significantly, as does the base current. The decrease in the current gain can be explained by changes in the effective carrier 76  3.3. DC Analysis lifetimes in the base. The quantum capture and stimulated recombination in the QW are fast processes competing against carrier transit in the base; these deplete the base so the effective carrier lifetime in the base decreases.  60  Current Gain (β  DC  ,β  Optical Power (a. u.)  AC  )  50.6 40  βDC β  AC  20  0 0  1  2  3  4  I (mA)  5  6  7  8  Figure 3.4: Calculated βDC and βAC (left axis, blue curves) and the output power (right axis, green curve) as a function of the base current. Above threshold, the current gain decreases corresponding to a decrease in the minority carrier lifetime in the base. The simulated device has a threshold of IB,th ≈ 1.5 mA. To further verify this explanation we need to prove that the transit time across the base is not significantly affected by the addition of the QW in the base region. For this we calculate the base transit time for the two cases: a regular BJT and a TL. We use the general definition for the transit time of  77  3.4. AC Analysis the base of the BJT: τt =  dQB , dIc  (3.32)  where the QB is the stored charge in the base region. Applying equation (3.32) to the triangular distribution in the base of an n-p-n BJT yields τt = WB2 /2Dn [22]. Using the values in table 2.1, for the case of the base without QW, τt = 1.92ps. For the case of the base with QW, we can use the results of the Figure 3.3(a) to calculate equation (3.32) and find τt = 2.08ps (difference of 8.33%), showing that the QW does not change the transit time significantly.  3.4  AC Analysis  In the presence of small-signal sinusoidal modulation of the base or emitter currents, common-emitter or common-base configurations respectively, we can assume that the excess carrier distribution has the following form:  δn = δN0 + Re δnm ejωt ,  (3.33)  where δN0 is the steady-state solution of the diffusion equation, δnm is the amplitude of the AC component, and ω is the modulation angular frequency. Using equations (3.8) and (3.33), we obtain the small-signal diffusion equation: jωδnm = Dn  ∂ 2 δnm δnm − . ∂x2 τB  (3.34)  78  3.4. AC Analysis Solving equations (3.8)-(3.15) to find the small-signal emitter and collector current densities (je and jb ) in terms of jv.s. and nv.s. results in: ⎛ je = nv.s.  cosh2  qDn ⎝ sinh Ld  WB 2Ld  jb = 2nv.s.  qDn sinh Ld  +  WB 2Ld  ⎞ WB 2Ld  ⎠ + jv.s. cosh  sinh  WB 2Ld  WB 2Ld  + jv.s. cosh  WB 2Ld  ,  , (3.35)  (3.36)  where Ld is the modified frequency-dependant diffusion length defined as: L2d =  D n τB . 1 + jωτB  (3.37)  At high frequencies diffusion can be a limiting factor, as the diffusion length decreases with frequency and carriers may recombine before reaching the QW. In typical QW the diffusion limit (∼ 80 GHz) is well above the intrinsic and practical factors [48]. It is interesting to note that the form of equation (3.36) is the same as the equation for the SCH laser. For the SCH laser the equation relating the input current density, j, to the virtual states current density and concentration is  j = 2nv.s.  qDn sinh Ld  WB 2Ld  + jv.s. cosh  WB 2Ld  .  (3.38)  The only difference would be in the choice of the parameters. This difference is not inherent to the TL operation. From this result we may conclude that the small-signal behavior of the TL in common-emitter configuration is  79  3.4. AC Analysis the same as the conventional laser and that there is no physical mechanism for a bandwidth enhancement. Importantly, the resonance frequency of the laser is unaffected by the TL in common-emitter operation. This result is in direct contrast to what is published in [28], where the effect of the transit time of the transistor was incorrectly incorporated in the laser rate equations so that the small-signal modulation bandwidth of the TL in the commonemitter configuration was over-estimated. However, equation (3.35) shows that when the signal is applied to the emitter in the common-base configuration, we will obtain a response different from the conventional SCH laser – this will be shown in the simulation results. Furthermore, none of the modeling papers, e.g., [29], [28] have sought to differentiate between the common-base and common-emitter configurations, and all have discussed the only common-emitter configuration. In the small-signal regime the laser operation is described by the linearized rate equations [11]. Equations (3.16) and (3.17) in the small-signal regime are: jωs = (ΓG0 S0 ) nqw − jωnqw =  jqw − qd  1 − S0 ΓτP  s−  S0 τP  s,  1 + G0 S0 nqw , τS  (3.39)  (3.40)  where the S0 is the DC value of the photon concentration which can calculated from equation (3.31). Using equations (3.35), (3.36), (3.39), and (3.40), the small-signal modulation response for the TL can be obtained.  80  3.4. AC Analysis From equations (3.39) and (3.40), we can write:  H1 (jω) =  s (jω) = jqw (jω) ω 2 − jω  − ΓGqd0 S0 1 τS  + G0 S0 +  S0 τP  −  G 0 S0 τP  +  S0 τS τP  , (3.41)  H2 (jω) =  s (jω) ΓG0 S0 . = nqw (jω) jω + τSP0  (3.42)  H1 (jω) and H2 (jω) are the transfer functions describing the dynamics of the photons in terms of the QW current and the QW carrier concentration, respectively. Equation (3.41) models the modulation response for a conventional laser neglecting the effect of carrier transport. It describes the modulation of the output power with respect to the current entering the active region (QW). If we ignore the gain compression effect for a while, i.e., = 0, equation (3.42) describes that photons and active area carriers are 90◦ out of phase. When there is resonance in the photon density the carrier density, gets depleted. Gain compression brings both quantities slightly in-phase. By using equations (3.19) and (3.20), we can find the transfer functions relating the photons to the virtual states current and carrier concentration: H3 (jω) =  s (jω) = jv.s. (jω)  1 1+  τcap τS  + jτcap ω  H1 (jω) H4 (jω) =  τ  +  qd τcap jω + esc  1 τS  , (3.43)  H2 (jω)  s (jω) 1 . = τcap 1 τesc 1 nv.s. (jω) + τesc H2 (jω) qd H1 (jω)  (3.44)  81  3.4. AC Analysis Finally, using the above results and equations (3.35) and (3.36), we can find the common-emitter and common-base modulation transfer functions: HCE (jω) =  s (jω) = jb (jω)  1 2 qDn sinh H4 (jω) Ld  WB 2Ld  1 + cosh H3 (jω)  WB 2Ld  ,  (3.45)  HCB (jω) =  s (jω) = je (jω)  1  ⎛ ⎜ n ⎜ 2 qD Ld ⎝sinh  WB 2Ld  cosh2 + sinh  H4 (jω)  WB 2Ld WB 2Ld  ⎞  .  ⎟ ⎟ ⎠  WB 2Ld H3 (jω)  cosh +  (3.46) The common-emitter and common-bas modulation responses, equations (3.45) and (3.46), will be studied in the following sections. Equations (3.45) and (3.46) are the photon concentration modulation response. In the following sections we will use the output power modulation response as in equation (2.11).  3.4.1  Common-Emitter Configuration  In this section we study the AC behavior of the common-emitter configuration. In common-emitter configuration as shown in the Figure 3.5, the base terminal of the TL serves as the input, the collector terminal is the electrical output and the emitter is common for both. The optical output is the light coming out of the base region. A BJT common-emitter amplifier is used in electric circuits as a high  82  3.4. AC Analysis gain stage. For the circuit shown in Figure 3.5, ignoring the laser operation for a while, for the current gain and voltage gain we have:  RL vo  iC  Collector Base  Bias-T  Laser Light  V CE  iB  RF  Emitter V BE  iE  Figure 3.5: An n-p-n TL in the common-emitter configuration. The baseemitter junction is forward-biased through VBE and the base-collector junction is reversed-biased through VCE . The base terminal is the electrical input and an RF signal is applied by using a Bias-T.  Ai Av  ic =β, ib  (3.47)  vo = −gm RL , vb  (3.48)  where Ai is the current gain, ic is the amplitude of the collector smallsignal current, ib is the amplitude of the base small-signal current, vo is the amplitude of the collector small-signal voltage, vb is the amplitude of the base small-signal voltage, RL is the load resistance connected to the collector, and gm is the transistor transconductance which is:  gm =  IC , VT  (3.49)  83  3.4. AC Analysis where IC is the collector bias current, VT is the thermal voltage which is: kB T where, kB is Boltzmann’s constant, T is the temperature, and q VT = q is the electron charge. With the laser operation we are mainly interested in the optical output. Figure 3.6 shows the normalized small-signal modulation response for the TL in the common-emitter configuration. This includes: (1) |H1 (jω)|2 =|s/iqw |2 :  Transfer Function (dB)  5 0  H  H1(jω)  (jω)  CE  −5  −10 H  (jω)  con.  −15  H (jω) 3  −20 −25 −30 0  17.2 GHz  39 GHz  41.5 GHz  20 40 Frequency (GHz)  60  Figure 3.6: Normalized small-signal modulation responses for the transistor laser, with respect to the different device currents. The intrinsic response has a bandwidth of 41.5 GHz (blue curve, labeled H1 (jω)); the quantum capture and escape effects slightly reduce the bandwidth to 39 GHz, green curve, labeled H3 (jω); and the overall response shows bandwidth of 17.2 GHz (red curve, labeled HCE (jω)). The reduction in bandwidth is due to the low-pass effect of the combination of the diffusion, quantum capture and escape processes. The modulation response of the SCH laser is labeled as Hcon. (jω). Both lasers are biased at IB = 5IB,th . the intrinsic response without quantum capture and diffusion effects, (2) |H3 (jω)|2 =|s/iv.s. |2 : the response including the quantum capture and escape effects, and (3) |HCE (jω)|2 =|s/ib |2 : the full response including the quantum 84  3.4. AC Analysis and diffusion effects. Similar to a SCH laser, the modulation response of the TL in the common-emitter shows parasitic effects which reduce the 3 dB modulation bandwidth. For comparison Figure 3.6 shows the modulation response of the SCH laser. Both lasers are biased at IB = 5IB,th . From the figure, both lasers show almost the same small-signal modulation behavior, the minor differences are due to using different values for the parameters.  3.4.2  Common-Base Configuration  In this section we study the AC behavior of the common-base configuration. In the common-base configuration as shown in the Figure 3.7, the emitter terminal of the TL serves as the input, the collector terminal is the electrical output and the base is common for both. The optical output is the light coming out of the base region. Emitter Laser Light Collector iE  iC  Base  vo  iB  Bias-T  V BE  RL  VCB  RF  Figure 3.7: An n-p-n TL in the common-base configuration. The Baseemitter junction is forward-biased through VBE < 0 and the base-collector junction is reversed-biased through VCB . The emitter terminal is the electrical input and an RF signal is applied by using a Bias-T. A BJT common-base amplifier is used in electric circuits as a high bandwidth stage. For the circuit shown in Figure 3.7, ignoring the laser operation 85  3.4. AC Analysis for a while, for the current gain and voltage gain we have: ic ≈1, ie  (3.50)  vo = g m RL , ve  (3.51)  Ai  Av  where Ai is the current gain, ic is the amplitude of the collector smallsignal current, ie is the amplitude of the emitter small-signal current, vo is the amplitude of the collector small-signal voltage, ve is the amplitude of the emitter small-signal voltage, RL is the load resistance connected to the collector, and gm is the transistor transconductance. Figure 3.8 shows the small-signal modulation responses of the commonbase and common-emitter configurations at different bias currents. There  Transfer Function (dB)  −10  IB=1.1IB,th=1.6 mA  Common Emitter  −20 −30 Common Base −40  IB=4IB,th=6 mA  IB=4IB,th=6 mA  −50 IB=1.1IB,th=1.6mA  −60 −70 1  Δ I =1.1 mA B  10 Frequency (GHz)  100  Figure 3.8: Transfer function for the small-signal modulation of the transistor laser considering both the common-emitter and common-base configurations. The laser threshold is IB,th = 1.25 mA. The bias current for the both configurations is varied from IB = 1.1IB,th to IB = 4IB,th with current step of ΔIB = 1.1 mA.  86  3.4. AC Analysis are three main differences between common-emitter and common-base modulation responses. First is the slope of the curves at frequencies above the resonance: the common-emitter configuration has a -60 dB/dec slope while the common-base configuration shows a slope of -40 dB/dec. The second difference is the low-frequency gain value: the common-emitter configuration has a higher DC gain than the common-base configuration, and finally in the common-emitter configuration curves, as the bias current increases, the damping increases and the peak corresponding to the relaxation oscillation frequency diminishes; at high bias currents, over-damping due to the gain compression and carrier transport effects are the limiting factors which saturate the bandwidth. In the common-base configuration, the peak corresponding to the relaxation oscillation is critically damped and the curves are flat; most importantly there is no bandwidth saturation effect in the small-signal response of the common-base configuration. The common-collector is another transistor configuration as shown in Figure 1.6(c). However, it turns out that its optical modulation response is exactly same as the common-emitter configuration. This is because the photons do not sense if either the emitter or the collector is grounded. To further investigate the physical origin of the removal of the gain compression of the common-base modulation response, we plot the intrinsic modulation response, i.e., |H1 (jω)|2 =|s/iqw |2 . This response does not include the quantum capture and diffusion effects and is obtained from equations (3.39) and (3.40). It has been shown in Figure 3.9. The relaxation oscillation peaks are apparent even at high bias. Next, we investigate the relationship between the current entering the 87  3.4. AC Analysis 0 Transfer Function (dB)  −10  IB=1.1IB,th=1.6 mA IB=4IB,th=6 mA  −20 −30 −40 −50 −60 −70 1  Δ I =1.1 mA B  −60 dB/dec 10 Frequency (GHz)  100  Figure 3.9: Intrinsic modulation response of the TL, s/iqw , without any transistor carrier dynamics (diffusion or quantum capture/escape). The bias current is varied from IB = 1.1IB,th to IB = 4IB,th . laser QW, and the applied external emitter current, Figure 3.10. In a conventional SCH laser and common-emitter, Figure 3.10(a), this corresponds to a low-pass filter parasitic due to the diffusion and quantum capture/escape effects (which reduce the bandwidth as shown in Figure 3.6). However, the situation is very different for the common-base configuration of the TL, Figure 3.10(b), where a band-reject filter is observed at exactly the resonance frequency of the intrinsic laser response, as well as a low-pass filter at approximately 100 GHz. The physical interpretation for this phenomenon is that during a relaxation oscillation, the excess electrons in the QW can escape from the laser and be swept out to the collector. This effectively dampens the relaxation oscillation. This is evident in Figure 3.10, which shows that the transistor adds an electrical equalization which removes the laser relaxation oscillation.  88  3.4. AC Analysis  I =4I B  −5 −10 −15 −20 1  I =1.1I B  =6 mA  B,th  =1.6 mA  B,th  Δ IB=1.1 mA 10 Frequency (GHz)  (a)  100  −10  −30  −20  qw e  0  /i , dB)  0  Current Ratio (i  qw b  Current ratio (i /i , dB)  5  −40 −50  IB=4IB,th=6 mA I =1.1I B  =1.6 mA  B,th  Δ I =1.1 mA B  −60 −70 1  10 Frequency (GHz)  100  (b)  Figure 3.10: Plot of the current ratios a) iqw /ib , this corresponds to the current entering the QW for a given applied base current. The electrical equalization effect of the transistor laser in the common-base configuration is evident, and removes the relaxation oscillation peak in the overall modulation response. b) iqw /ie , this corresponds to the current entering the QW for a given applied emitter current. The low-pass effect is evident. The frequency dependant current gain (β) of the transistor laser shows a resonance effect, which is shown in Figure 3.11. The increased transistor β at the laser resonance frequency indicates that more carriers are being swept out to the collector, while less photons are emitted at the resonance. The damping of the relaxation oscillation is also partially evident in the common-emitter case, where the oscillation peaks are reduced in magnitude. The physical interpretation is again that the electrons participating in the relaxation oscillations can escape the QW and be swept out to the collector. We note that the relaxation oscillations are not completed removed. The experimental measurements in Ref. [26] show that the modulation responses have damped resonance peaks with a magnitude of approximately 1 - 2 dB, with a bandwidth below 20 GHz, in qualitative agreement with this model.  89  3.4. AC Analysis 35 30 IB=1.1IB,th=1.6 mA  c b  β=i /i (dB)  25  Δ I =1.1 mA B  20 15 10  I =4I =6 mA B B,th  5 0 1  10 f (GHz)  100  Figure 3.11: Small-signal current gain of the transistor, 20 log β, in dB. The laser relaxation oscillations alter the behavior of the transistor, giving rise to higher current gains near the laser resonance. The bias current is varied from IB = 1.1IB,th to IB = 4IB,th . By decreasing the capture time, or increasing the escape time, or increasing the length of the base, we can artificially reduce the effect of the transistor on the transistor laser overall response. In such a case, we find the modulation responses of the common-emitter and common-base configurations approach the intrinsic laser response (but with a difference in RF gain due to the transistor beta), shown in Figure 3.12. The relaxation oscillation peaks increase in magnitude for both common-emitter and common-base configurations. To illustrate the gain compression removal in the common-base configuration, we show the bandwidth variation of the common-emitter and common-base configurations as a function of the bias current (IB ) in Figure 3.13. For comparison, the intrinsic modulation bandwidth is also shown. At low bias currents both configurations show almost the same bandwidth. 90  3.4. AC Analysis 0  Transfer Function (dB)  I =1.1I B  −10  =0.6 mA  B,th  Common Emitter IB=4IB,th=2.3 mA  −20 −30 −40 Common Base −50 −60 −70 1  IB=1.1IB,th=0.6mA Δ I =0.4 mA B  10 f (GHz)  100  Figure 3.12: Transfer function for the small-signal modulation of a transistor laser, where the effect of the transistor has been reduced by increasing the escape lifetime by a factor of 10X (threshold is thus reduced to IB,th = 0.5mA). The result is that the response becomes more similar to the intrinsic laser response, with stronger relaxation oscillations, and lower modulation bandwidths. The bias current for the both configurations is varied from IB = 1.1IB,th to IB = 4IB,th . As IB increases the bandwidth of the common-base configuration increases rapidly while the bandwidth of the common-emitter configuration saturates, as in the case of a regular SCH laser which is discussed in Chapter 2 and [48]. The different modulation bandwidths seen in the TL configurations are due to the different types of carriers used in the common-emitter and commonbase configurations, i.e., holes in the common-emitter case and electrons in the common-base for an n-p-n TL. This is shown in the Figure 3.14. Figure 3.14(a) shows what happens in the common-base modulation. The injected electrons from the emitter to the base have three options: 1) direct recombination with the base valence band holes, 2) capturing into the QW and 91  3.4. AC Analysis 60  Bandwidth (GHz)  50 Intrinsic 40  Common Base  I =1.1I B  B,th  I =10I B  30  B,th  20 Common Emitter, SCH  10 0 0  2  4  6  8 10 I (mA)  12  14  B  Figure 3.13: Bandwidth variation of the transistor laser and intrinsic response, versus DC bias current. The upper (red) curve shows the intrinsic bandwidth of the device. This curve does not incude any parasitic effects and represents the upper limit for conventional lasers’ bandwidth. The middle (blue) curve shows the bandwidth of the common-base configuration. The bandwidth of the common-base closely follows the intrinsic bandwidth but is smaller due to low-pass filtering effect of the carrier dynamics parasitic. A bandwidth of 48 GHz is predicted for 10IB,th . The lower (green) curve shows the bandwidth of common-emitter configuration, which is also identical to the SCH conventional laser case. A maximum bandwidth of 17 GHz is observed. The bias current for all curves is varied from IB = 1.1IB,th to IB = 10IB,th . recombination with the QW valence band holes, and 3) getting collected by the collector junction. The injected electron average lifetime is affected by the above processes which are characterized by the lifetimes τB , τS , and τt , respectively. Figure 3.14(b) shows what happens in the common-emitter modulation. The injected holes from base terminal to the base have two options: 1) direct recombination with the base conduction band electrons, 2) capturing into the QW and recombination with the QW conduction band  92  3.4. AC Analysis  1 E F,E  2  4 EF,B  3  3  E F,E  EF,B  1 2 EF,C  EF,C  Base Terminal  (a)  (b)  Figure 3.14: Energy band diagram of the an n-p-n TL in the normal mode: emitter junction is forward-biased and collector junction is reversed-biased. a) Common-base modulation. The injected electrons to the base through the emitter junction have three choices: direct recombination with the base valence band holes (path 1), capturing to QW and recombination with the QW valence band holes (paths 2 and 3), or getting collected by the collector junction. b) Common-emitter modulation. The injected holes to the base through the base terminal have two choices: direct recombination with the base conduction band electrons (path 1), capturing to QW and recombination with the QW conduction band electrons (paths 2 and 3). electrons. The injected hole average lifetime is affected by the above processes which are characterized by the lifetimes τB and τS , respectively. The above argument shows that different types of carriers have different lifetimes. From simple 1-level rate equations (2.1) and (2.4), we can show that [1]:  ωr =  1 + G0 τP ΓNtr τ τP  I0 −1 , Ith  (3.52)  where, τ is the carrier lifetime, Ith is the threshold current. From equation (3.52), ωr is inversely proportional to the square root of τ . Decreasing the lifetime τ increases the relaxation oscillation frequency ωr and band bandwidth increases. Alternatively, the difference between the modulation response if the 93  3.4. AC Analysis common-emitter and common-base can be understood from transistor charge control analysis [22]: iE =  QB dQB + , τt dt  (3.53)  iB =  QB dQB + , τB dt  (3.54)  where QB is the stored charge in the base region when the transistor is in its normal mode. From these equations the small-signal base current is related to stored charge in the base region through the carrier recombination lifetime (τB ), while the small-signal emitter current is dependent on the transit time (τt ), which is in the range of pico-seconds. In TL case two mechanisms, carrier transit through the base and stimulated emission work in parallel to reduce the carrier effective lifetime in the base region, thereby enhancing the small-signal bandwidth in the common-base case. For moderate bias levels, the modulation response of the common-base case approaches the intrinsic modulation response. This indicates that if a laser can be designed to have a large intrinsic modulation bandwidth with a high resonance frequency, then the common-base transistor laser structure is an effective method of improving the modulation bandwidth by removing the damping effects due to the carrier dynamics, thereby ultimately realizing nearly the full inherent bandwidth. It should be noted that the bandwidth cannot be increased beyond the intrinsic bandwidth of the laser, compared with the injection locking mechanism where the bandwidth is enhanced beyond the intrinsic bandwidth of the laser [55].  94  3.4. AC Analysis  3.4.3  3rd Order Modulation Response Approximation  In this section we present a polynomial transfer function for the small-signal modulation response of the common-emitter and common-base configurations. We represent both modulation responses by:  H(f ) =  1 f 1+j f0  ·  1 f γ f 1 − ( )2 + j · fr 2πfr fr  ,  (3.55)  where f0 is parasitic pole frequency due to diffusion, transit through the base, quantum capture and escape lifetimes; γ is the damping rate; fr is the effective resonance frequency of the modulation response. Our simulations show that a 3rd order response, equation (3.55), can accurately fit the analytic modeling results, as shown in Figure 3.15. To compare the common-base and common-emitter modulation responses, we perform the curve fitting of equation (3.55) to the common-base and common-emitter responses obtained through the numerical modeling in the previous sections.  Table 3.2 shows the parameter values obtained after curve fitting using equation (3.55) to the modeling results at two different bias points IB = 2IB,th and IB = 3IB,th . The parasitic pole of the common-base configuration is much larger than the common-emitter configuration parasitic pole, so in frequencies < 100 GHz the common-base shows a −40 dB/dec rate. One important point is that equation (3.55) predicts a −60 dB/dec roll-off for both common-base and common-emitter configurations for very high frequencies. However, the value of the parasitic f0 is very different  95  3.4. AC Analysis  Transfer Function (dB)  5 Common−Base  0 −5 −10 −15  Common−Emitter  −20 −25 −30 −35 1  28 GHz 17 GHz 10 Frequency (GHz)  100  Figure 3.15: Normalized transfer functions for the small-signal modulation of the transistor laser in common-emitter and common-base configurations, at IB = 3IB,th . Solid curves show the curve-fit by equation (3.55), while the circles are the data points from the simulated results. for common-base and common-emitter configurations (> 100 GHz vs. < 30 GHz, respectively). Thus, in the practical range of frequencies (< 100 GHz), the common-base has a −60 dB/dec slope while the common-emitter shows a −40 dB/dec slope. The curve-fitting also indicates that the effective resonance frequency of the common-base is enhanced, owing to the equalization effects which remove the relaxation oscillation peak and leaves a critically damped overall modulation response. There is very good agreement between the two curves showing that the third order transfer function described by equation (3.55) models the small-signal modulation response of the TL in common-base and common-emitter configurations.  96  3.5. Feedback in TL Parameter fr  γ f0  Common-emitter 10, 16.5 60, 124 29, 26  Common-base 16, 23 134, 181 108, 108  Unit GHz Hz GHz  Table 3.2: Equivalent 3rd order model parameter values found by curvefitting. The two numbers in each row correspond to values for IB = {2IB,th , 3IB,th }.  3.5  Feedback in TL∗  By using feedback we have more freedom to exploit the gain-bandwidth trade-off in the TL modulation. Additional equations for the voltage-current relations at each terminal are added by using the law of the junction [22]:  n(0) =  n2i qVBE /kB T e , NB  n(WB ) = 0 ,  (3.56)  (3.57)  where n(0) is the carrier concentration at emitter-base interface, ni is the intrinsic carrier concentration, NB is the base doping, VBE is the applied base emitter voltage, kB is Boltzmann’s constant, T is the temperature, and the quasi-neutral base lies between x = 0 and x = WB . Figure 3.16 shows the transistor laser with the capacitive feedback (Cf ) from the collector to the base. Cf is assumed to be a short in AC operation. This is an example of the current-current feedback [27], meaning that the feed-forward amplifier ∗ A version of this section has been published as a conference paper: B. Faraji, N. A. F. Jaeger, L. Chrostowski, “Modelling the Effect of the Feedback on the Small Signal Modulation of the Transistor Laser,” Photonics Society Annual Meeting, Denver, Colorado, US, 2010.  97  3.5. Feedback in TL  Figure 3.16: Simulation circuit. The circuit shows the necessary elements to bias and modulate the TL with capacitive feedback. Inductors (Lb and Lc ) are used for the DC biasing, capacitor Cb is used to inject RF signal to the base, and capacitor Cf is used as a feedback from collector to the base. Lb and Lc are assume short, zero resistance, in the DC operation and open, infinite resistance, in the AC operation. Cb is assumed open, infinite resistance, in DC operation and short, zero resistance, in the AC operation. Cf = 50 pF is used in the simulations. is characterized by a current gain βac (ω) and the feedback network by a current ratio. The RF signal is injected through the coupling capacitor Cb , and inductors Lb and Lc are used to bias the device. With this circuit, the laser transfer function is changed. Without feedback the small-signal modulation transfer function, H(ω), is defined as:  H(ω) =  s(ω) . ib (ω)  (3.58)  In fact H(ω) is the transfer function of the TL without feedback in the common-emitter configuration. While with feedback the transfer function,  98  3.5. Feedback in TL Hf (ω), is defined as: Hf (ω) =  s(ω) , im  (3.59)  where Hf (ω) is transfer function with the feedback and im is the amplitude of the RF current injected to the base of the TL. Writing Kirchhoff’s current law in the base node we get:  im = ib + ic = ib (1 + βac (ω)) .  (3.60)  Substituting im from equation (3.60) into equation (3.59) and using equation (3.58), results in: Hf (ω) =  H(ω) . 1 + βac (ω)  (3.61)  From equation (3.61) it is obvious that the transfer function of the laser is modified by using the feedback and, as we showed in [56], in a transistor laser, the quantum well carriers are coupled to the excess minority carriers in the base; oscillations in the quantum well carriers thus lead to oscillations in the base minority carriers, and to oscillations in the collector current. Hence, the relaxation oscillations seen in the optical output are also evident in the collector current. This is reflected in the frequency response of the transistor. By using the feedback and injecting collector current to the base we can effectively remove the resonance in the laser transfer function. Ideal capacitive feedback, i.e., direct base-collector connection, is exactly the same as in common-base configuration. It is may be possible to engineer the smallsignal modulation response of the TL by utilizing more elaborate feedback 99  3.5. Feedback in TL systems. Figure 3.17(a) shows the small-signal modulation response of the commonemitter configuration without feedback while Figure 3.17 (b) shows the modulation response with feedback. The TL with feedback shows very similar behavior to the common-base configuration, i.e., the DC gain decreases and bandwidth increases (bandwidth equalization), a reduction in damping at a low bias current and -40 dB/dec slope in high frequencies. −30  −20 9 mA IB  −40 −60  Δ I = 1.8 mA  1.8 mA  B  −80 −100 0.1  −60 dB/dec 1 10 Frequency (GHz)  (a)  100  Transfer Function (dB)  Transfer Function (dB)  0  −40  9 mA IB  −50 −60  Δ I = 1.8 mA  1.8 mA  B  −70 −40 dB/dec  −80 −90 0.1  1 10 Frequency (GHz)  100  (b)  Figure 3.17: Small-signal modulation transfer function of the TL. a) Common-emitter configuration without feedback. Bandwidth varies from 11.4 GHz to 18.5 GHz. b) Common-emitter configuration with feedback. Bandwidth varies from 12.7 GHz to 36.4 GHz. The bias current is varied from IB = 2IB,th = 1.9 mA to IB = 10IB,th = 9 mA We can analytically compare the common-emitter configuration with an ideal capacitive feedback with the common-base configuration. The transfer  100  3.5. Feedback in TL function of the common-base configuration can be written as:  Hcb (ω) = =  s(ω) ie s(ω) ib (1 + β(ω))  = Hf (ω) .  (3.62)  Ideally, the common-emitter with a capacitive feedback will have the same characteristics as common-base configuration. However, due to limited response of the feedback network common-base and common-emitter with feedback will have different responses. To elaborate this, suppose the electric feedback has an input-output relation as:  iout = Vf (ω)iin ,  (3.63)  where iout is the output current, iin is the input current, and Vf (ω) is the feedback transfer function. Using Equation (3.63) in Equation (3.61), we can write the the optical transfer function of the TL as:  Hf (ω) =  H(ω) . 1 + Vf (ω)βac (ω)  (3.64)  101  3.6. Model Verification  3.6  Model Verification∗  In this section we verify some aspects of the developed model with the simulation results obtained from a commercial package. We have used an advanced numerical simulation software (Crosslight PICS3D) [57] that solves the electrical and optical models self-consistently. The carrier transport is described based on the classic drift-diffusion model [58], with the thermionicemission model used at hetero-junctions. The QW capture and escape processes are described by phonon-scattering theory. Lateral optical modes are calculated by the effective-index method [59]. In the QWs, the conduction bands are assumed to be parabolic, and the valence bands are calculated by the 6×6 kp method for the valance-band mixing [60]. Coulomb enhancement (many-body effect) [61] is involved in calculating optical gain. The electrical and optical models are solved self-consistently by the finite-element method and Newton’s method [57]. To verify the software models, we have simulated a conventional VCSEL that has the same QW structure, taking self-heating effect into consideration. The simulated results from PICS3D and experimental data are demonstrated in Figure 3.18. There is a good agreement between the measurements and simulations. In the next step a transistor VCSEL has been designed; Figure 3.19 shows the structure [62]. It is an N-p-n In0.49 Ga0.51 P / GaAs HBT structure. The bottom and top distributed Bragg reflectors consist of 30 pairs and 24 ∗  A version of this section has been published as an invited journal paper: W. Shi, B. Faraji, M. Greenberg, J. Berggren, Y. Xiang, M. Hammar, M. Lestrade, Z. Li, S. Li, L. Chrostowski, “Design and modeling of a transistor vertical-cavity surface-emitting laser,” Optical and Quantum Electronics, 1 - 8, 2011 (Invited).  102  3.6. Model Verification 3  3 Experiment 2.5  Simulation  2  1.5  1.5  1 Experiment Simulation  Voltage  2  0.8  1  Intensity, a.u.  Optical power, mW  2.5  0.5  0.4  0.5  0.2  0 900  0 0  1  0.6  2  4  6 Current, mA  920  940 960 Wavelength, nm  8  10  980  1000  0 12  Figure 3.18: PICS3D simulation and measurement data. LIV curves of a MQW In0.17 Ga0.83 As / GaAs VCSEL. The photoluminescence data for the simulation and experiments are shown in the inset. pairs of Al0.85 Ga0.15 As / GaAs, respectively. The base region utilizes an asymmetric base doping profile where the whole base region is composed of a 15 nm heavily doped (1×1019 cm−3 ) layer, a 30 nm doping grading layer, three intrinsic In0.17 Ga0.83 As / GaAs QWs, another 30 nm doping grading layer, and a 40 nm heavily doped (1×1019 cm−3 ) base contact layer. The heavily doped layers are aligned with the valleys of the longitudinal standing wave in the vertical optical cavity to reduce the optical absorption. A 6 μm oxide aperture is used in the simulation [62].  103  3.6. Model Verification E contact DBR  n AlGaAs/GaAs  Oxidation Spacer Emitter Top base MQW Bottom Base Collector Sub-collector  Al0.98Ga0.02As AlGaAs n In0.49Ga0.51P p+ GaAs i InGaAs/GaAs p+ GaAs i GaAs n GaAs  DBR  n AlGaAs/GaAs  B contact  C contact  Figure 3.19: Structure of transistor VCSEL. It is an N-p-n InGaP/GaAs HBT with 30 and 24 pairs of AlGaAs/GaAs layers as bottom and top DBRs, respectively. By simulating the structure of Figure 3.19, we find the LI curve of the device, Figure 3.19(a). Also transistor terminal current characteristics (IC vs IB ) are shown in the Figure 3.19(b). The average gain function, G(N ), of one QW is extracted from the PICS3D simulation. It is assumed that all the QWs contribute equally to total gain. The gain function is multiplied by number of the QWs in the design and is then used in our model. We obtain good agreement between the PICS3D and our model after fine-tuning the parameters. Our model results are super-imposed on the PICS3D results in Figure 3.19. Table 3.3 shows the parameters values extracted from the PICS3D simulations. For comparison we repeat the parameter values used in our previous simulations. The effective QW width used in our simulations to fir our model results to numerical simulations is 3 (number of the QWs used in  104  3.6. Model Verification  Analytical Model PICS3D  60 50  1  IC (mA)  Optical Power (mW)  1.5  40 30 Analytical Model PICS3D  20  0.5  10  0 0  2  4 6 Base Current (mA)  8  10  0 0  (c)  2  4 6 IB (mA)  8  10  (d)  Figure 3.19: a) The LI curve of the TVCSEL. The results of PICS3D and our developed model are shown. b) IC vs IB . Above laser threshold the current gain of the transistor drops, it is predicted by both simulations: PICS3D and our model. Figure 3.19) times the single QW thickness. By using the parameters, we then simulate the small-signal modulation response of the designed TVCSEL in common-emitter and common-base configurations in the way described in Section 3.4. Figure 3.20 shows the results. A few observations can be seen from this figure: 1) DC gain of the common-emitter is higher than the common-base, 2) common-base has a slope -40 dB/dec at high frequencies while it is -60 dB/dec for the commonemitter, 3) the peak in the transfer function is shifted to the higher frequencies in common-base configuration. A bandwidth of 25 GHz for the common-base configuration at bias current of IB = 8 mA is predicted.  105  3.7. Discussion Parameter (symbol) A τS τP τcap τesc τB Dn vg αm αi d Γ β  Extracted 20π 300 5 1.9 30 150 36 9.6 45 20 3×6.7 0.04 10−4  Simulation 9π 500 3 1 20 200 26 5.13 40 25 12 0.05 10−4  Unit μm2 ps ps ps ps ps cm2 /s cm/ns cm−1 cm−1 nm -  Table 3.3: Values of the parameters. The first column shows the parameters, the second column is the value obtained by comparing our model result with the numerical simulation. The third column shows the values we used in our model. The values are chosen for a 980 nm TVCSEL.  3.7  Discussion  In this section we discuss some of the limitations and extensions to this model.  3.7.1  Trade-off between RF Gain and Bandwidth  The most important limitation in a laser is due to the low-pass nature of the frequency response. This would ideally be overcome by an increased resonance frequency with a critical damping rate, with no change in the RF amplitude. Unfortunately, in the transistor laser, we have shown that there is no resonance frequency enhancement in the common-emitter configuration, as compared with a conventional laser. However, we have shown that the common-base configuration shows a larger bandwidth, with a trade-off 106  3.7. Discussion  Transfer Function (dB)  0 −10 −20 −30 −40  Δ IB=2 mA  IB=2 mA IB=2 mA  I =8 mA B I =8 mA B −60 dB/dec  −50 −60 −70 1  Common Emitter Common Base −40 dB/dec 10 Frequency (GHz)  40  Figure 3.20: Transfer function for the small-signal modulation of the transistor laser considering both the common-emitter and common-base configurations. Bandwidth varies from 6 GHz to 18.5 GHz for common-emitter. Bandwidth varies from 6.5 GHz to 25 GHz. The laser threshold is IB,th = 1.5 mA. The bias current for the both configurations is varied from IB = 2 mA to IB = 8 mA with current step of ΔIB = 2 mA. in RF gain. Hence, there may be some potential for applications to utilize the common-base configuration if the RF gain can be improved. The trade-off between gain and bandwidth has been exploited in many devices (e.g., feedback in transistors) to obtain a larger modulation bandwidth. In order to obtain a large modulation bandwidth in a conventional laser, an equalizer and an amplifier could be employed. Specifically, this could be in the form an electrical filter at the input, or an optical filter at the output [63]. In both configurations, there are some challenges, namely 1) the RF phase shifts by 180◦ beyond the resonance frequency, hence the filter would need to adjust the phase, 2) the equalization would introduce additional noise, and 3) additional components are needed. In the transistor 107  3.7. Discussion laser operated in common-base, the equalization is done internally in the device, hence all that is required is a flat frequency response current amplifier at the input. This can be achieved by adding a wide-bandwidth amplifier stage at the emitter (the cascode configuration) [27] to compensate the gain reduction of the common-base configuration. Figure 3.21 shows such a design. The idea is to introduce a high small-signal resistance in the emitter of Q2 . 2  Laser Light  L  1 i RF = i msinȦt  Figure 3.21: Cascode configuration. Q2 is the TL while Q1 is the driving transistor acting as a high output resistance current source. The analysis of the circuit shown in Figure 3.21 can be summarized as follows. Transistor Q1 is in the common-emitter configuration and its current gain according to the equation (3.47) is:  Ai (ω) =  ic1 = β(ω) , im  (3.65)  where im is the amplitude of the RF current source, β(ω) is the base to collector gain amplification factor of the transistor Q1 . TL Q2 is in the common-base configuration and its optical transfer function is found by  108  3.7. Discussion using equation 3.46:  HQ2 (ω) =  s(ω) = HCB (ω) , ie2  (3.66)  where s(ω) is the AC part of the photon concentration. In the figure 3.21, ie2 = ic1 , and the total modulation response of the cascode configuration is:  Hcascode (ω) =  s(ω) = β(ω)HCB (ω) . im  (3.67)  The gain of the transfer function comes from β(ω) and the bandwidth comes from the HCB (ω).  3.7.2  Modeling the Multi-Quantum Well (MQW) TL  In Section 3.2.1 we assumed that there is just one QW in the middle of the base region. This assumption is made to simplify the mathematical derivations. In practical cases, for conventional lasers one QW is rarely used in laser design due to insufficient optical gain. From our results in Section 3.6, single QW system can be used to model a MQW system, however, it parameters need to be modified (e.g., gain, thickness). Single QW model does not provide detail information on the carrier dynamics of the MOQW system. To more accurately model a MQW TL, we need to modify the model to include: 1) multiple quantum capture and escape of the carriers in the transport across the base, 2) tunnelling of the 2D QW carriers to the adjacent wells. Multiple quantum capture and escape of the carriers can be  109  3.7. Discussion addressed by solving the continuity equation, equation (3.6), for each region between the QW, and link the current and carrier concentration of the virtual states of each QW to the QW 2D carriers through the quantum capture and escape lifetimes (τcap and τesc ). Tunneling of the carriers from a QW to the neighbor QWs can be addressed by using the proper tunneling lifetime (τtun ) which can be calculated for given QW parameters (QW thickness, energy barrier height and thickness). We may modify the rate equations for the QWs carriers, equation (3.17), to include the tunneling effect. When there is more than one QW, solving the diffusion equation with the rate equations will result in very large equations which may not be easy to solve analytically. One possibility is solving the equations by a numerical method like the finite difference method.  3.7.3  Limitation of the Diffusion Model  In the modeling, we assumed that diffusion is the dominant mechanism for transport across the base and we neglected other mechanisms (e.g., drift of carriers). This assumption may be violated when the base thickness is comparable to the carriers mean free path in base region. In a thin base BJT, the ballistic current becomes dominant. We need to update the equations and the boundary conditions accordingly [64]. The diffusion assumption can be used safely when the carrier concentration changes slowly in a distance equal to a mean free path [54]. In HBTs, the carriers injected to the base will gain kinetic energy because of the discontinuity in the emitter and base conduction band. The hetero-junction step acts like an electric field impulse which accelerates the 110  3.7. Discussion carriers. These carriers in the thin base will have the probability statistics of ballistic transport. However, in the heavily doped base, the carrier-plasmon scattering rate is so high that the mean free path can be very low, and ballistic transport will be small [54]. Hence, for the HBT TL devices considered, the diffusion approximation is justified.  111  Chapter 4  Large-Signal Analysis 4.1  Introduction  Large-signal modulation is related to small-signal modulation [11], and in general, an improvement of the small-signal modulation response enhances the large-signal modulation response leading to a higher bit-rate. In large signal (digital) modulation, the frequency of the optical power oscillations in transition from one current level to another current level are closely related to the relaxation oscillation frequency [11]. Increasing the relaxation oscillation frequency reduces the rise time and increasing the damping (equivalent to the peak height in the small-signal modulation response) reduces the settling time of the large signal response. In short, having wide-band response for the small-signal modulation improves the large-signal modulation response. In large-signal modulation the deviation from the steady-state is comparable to the steady-state values themselves. The small-signal modulation results cannot be directly applied to study the large-signal modulation behavior. To model the large-signal dynamics, we need to solve the general form of the equations of the model in the previous chapters. These equations are valid for the large-signal modulation provided the nonlinear changes in 112  4.1. Introduction the gain with the carrier and photon density are included. The equations cannot be solved analytically, therefore we need to use the numerical technique developed in Chapter 2. In semiconductor lasers with the modulation of the electric current, both photon density and carrier density are modulated. Photon modulation was discussed in Chapter 2. Carrier modulation in the small-signal modulation regime can be found by solving rate equations 2.1 and 2.4:  nm (ω) = im (ω)  jω qdA ω − jω 2  1 + G0 S0 τs  G0 S0 − τP  .  (4.1)  With the carrier density modulation, the index of the active region, na , is modulated. As a result the cavity length is modulated causing the resonant mode to shift in frequency. This is called frequency modulation (FM) or frequency chirping [11]. This parasitic FM most of the time is undesired. In large-signal modulation because of the large variation in the electric current FM effects are bigger. FM results in the spectral broadening of the laser field which in the dispersive fibers increases spreading of the optical pulses. Pulse spreading results in inter-symbol interference which limits the performance of the digital optical links. FM is related to the laser line-width enhancement factor, α, by [36]: Δν =  α ΓG0 ΔN , 4π  (4.2)  where Δν is the frequency shift, ΔN is the carrier change because of the modulation. In this chapter we study the digital modulation performance of the TL in 113  4.2. Large-Signal Modulation different configurations and study the turn-on delay and FM performance.  4.2  Large-Signal Modulation  By using the model which was developed in Chapter 3 and solving the equations in the large-signal domain, we can find the large-signal behavior of the device. Figure 4.1 shows the variation of the output optical power and quantum well carrier density of a SCH laser. In these simulations we  10  2  5  P  out  (mW)  N  P  0 0  2  4  6  8  Carreir Density (1018 cm−3)  4  0 10  t (ns)  Figure 4.1: Turn-on delay for a SCH VCSEL. Laser threshold is Ith = 0.7 mA. Output optical power and carrier density variations are shown. have used the same values for parameters as in Chapter 3. The bias current at t = 0+ increases from zero to I(0+ ) = 2Ith . A few interesting effects take place during the transition time. Initially, the carrier density increases to fill up the reservoir, in the beginning, there is little optical power as the  114  4.2. Large-Signal Modulation spontaneous emission is the dominant process in producing the photons. As the carrier density reaches its threshold value, the stimulated emission increases and finally it limits further increase of the carriers. The time needed for the carrier density to reach its threshold is called turn-on delay, td . From Figure 4.1, td ≈ 1 ns. There are oscillations in both photon and carrier dynamics, the frequency of the oscillations is equal to the laser relaxation oscillation frequency at the new bias condition. However in the TL the situation is different and depending on the modulation configuration, the results would be different. Figure 4.2 shows the variation of the output optical power and quantum well carrier density of a TL in the common-emitter configuration. The base bias current at t = 0+ 2  2 −3  cm )  N 1.5  1  1  P  0.5  0 0  0.5  0.5  1 t (ns)  1.5  Carreir Density (10  Pout (mW)  18  1.5  0 2  Figure 4.2: Turn-on delay for a TL in common-emitter configuration. Laser threshold is IB,th = 1 mA. Output optical power and carrier density variations are shown. increases from zero to IB (0+ ) = 2IB,th . In Chapter 3, we showed that the 115  4.2. Large-Signal Modulation small-signal modulation response of the TL in the common-emitter configuration is exactly the same as the conventional SCH laser. This is the case for the large-signal as well. If we choose the same values for the simulation parameters, the results will be the same. For a TL with suitable parameters, from Figure 4.2, td ≈ 164 ps. The difference between the turn-on delays between SCH laser and TL in common-emitter configuration comes from the parameter values [24]. This difference is mainly due to the fact that the SCH layer is intrinsic while the base of the TL is highly doped. This leads to difference in the diffusion coefficient, Dn , recombination lifetime, τB , escaping lifetime, τesc , of the two structures. Figure 4.3 shows the variation of the output optical power and quantum well carrier density of a TL in the common-base configuration. At t = 0+ a step in current is applied, the laser is biased so that the optical output power at steady-state is the same as the common-emitter configuration. From the figure, the turn-on delay time equals 47 ps which is much less than the common-emitter configuration. The decrease in the turn-on delay can be explained by considering the different lifetimes involved in the common-emitter and common-base configurations. For an n-p-n TL as shown in Figure 3.14, in common-base configuration electrons in conduction band play important role in the modulation, while in the common-emitter configuration holes in the valence band are used for modulation. The Electron lifetime is smaller than the hole lifetime due to extra path to the reversed-biased collector junction. Mathematically, the effect of the transistor on the turn-on delay can be explained by the simple rate equations. Numerical analysis of the full rate 116  4.2. Large-Signal Modulation  2  2  1  1 P  0 0  0.5  1 t (ns)  1.5  Carreir Density (10  Pout (mW)  18  −3  cm )  N  0 2  Figure 4.3: Turn-on delay for a TL in common-base configuration. Laser threshold is IB,th = 1 mA. Output optical power and carrier density variations are shown. The emitter bias current at t = 0 increases from zero to IE (0+ ) = 46 mA. This value corresponds to the 2IB,th . equations [65] shows that if the laser is switched on from below threshold, the stimulated and spontaneous recombination terms in the photon rate equation are small and can be ignored. In this regime assuming there is no photon build-up below threshold, carrier density can be written as: dN N i − , = dt eVactive τ0  (4.3)  where time constant τ0 summarizes all the escaping processes from the active region, e.g., radiation and non-radiation recombination and quantum escape. Assuming this lifetime is constant an immediate solution of this first order differential equation for a step in the current (switching from 0 to i0 at 117  4.2. Large-Signal Modulation t = 0+ ) will be: t − i 0 τ0 τ (1 − e 0 ) . N (t) = eVactive  (4.4)  Because of the stimulated emission, the carrier density will be clamped at Nth . The time required for N to reach Nth defines the turn-on delay. Setting N (ttd ) = Nth and solving for td we obtain: td = τ0 ln  i0 . i0 − ith  (4.5)  Equation (4.5) shows the linear dependence of the turn-on delay on the carrier lifetime. As we explained in Chapter 3 for TL in the commonbase configuration the effective carrier lifetime in the base is reduced and common-base configuration shows better turn-on delay. However finding an analytical formula for the turn-on delay from the model is difficult because of the complexity of the describing equations, so we use the results obtained by numerical solution. The improved turn-on delay of the TL in common-base configuration can bring about better performance in digital modulation. Figure 4.4 shows the eye-diagram of the TL in the common-emitter and common-base configurations for bit rates of 2.5 Gbps and 10 Gbps. The input data stream used in simulation included 500 non-return zero (NRZ) pseudorandom square pulses with a finite rise-time and fall-time of 0.08 bit period and finite bandwidth. As in practical applications the rise-time of the input signal maybe significant due to the finite speed of the electronic drive circuit and the slowing-down effect of the parasitics. The input data stream range is cho-  118         Amplitude (AU)  Amplitude (AU))  4.2. Large-Signal Modulation   0  Eye height  í  Eye height 0 í  í í 0    0.2  0.4 t (ns)  0.6  í 0  0.8  0.2  1.5  1.5  1  1  0.5 0  Eye height  í í í 0  0.6  0.8  0.15  0.2  (b)  Amplitude (AU)  Amplitude (AU))  (a)  0.4 t (ns)  0.5 0  Eye height  í í  0.05  0.1 t (ns)  (c)  0.15  0.2  í 0  0.05  0.1 t (ns)  (d)  Figure 4.4: Eye-diagram of the digital modulation of the TL in the commonemitter and common-base configurations for different bit-rates. High level and low level are shown with red lines. The eye height is shown as the difference of the two 3σ points of high and low levels. a) 2.5 Gbps, commonemitter. b) 2.5 Gbps, common-base. c) 10 Gbps, common-emitter. d) 10 Gbps, common-base. The laser has threshold of IB,th = 1 mA. The amplitude of the data stream is varying from IB,min = 2 mA to IB,max = 4 mA and ER = 5 dB. Parameters of Table 3.1 are used in the simulations. The choice of IB,max = 4 mA comes from the fact that common-emitter has it maximum bandwidth, e.g. Figure 3.13  119  4.2. Large-Signal Modulation sen so that both common-emitter and common-base configurations give the same optical power in steady-state. To plot the eye-diagram the optical power is normalized as follows:  Pnorm =  Po − μ o , σo  (4.6)  where, μo is the average power of the output bit stream and σo is its standard deviation. Extinction ratio defined as:  ER = 10 log10  Po,max Po,min  ,  (4.7)  where ER is the extinction ration (dB), Po,max is the optical power in highlevel, and Po,min is the optical power in the low-level. Figure 4.5 shows the eye-diagram of the TL in the common-emitter and common-base configurations for bit rates of 20 Gbps and 40 Gbps. Comparing Figures 4.5 and 4.4 shows that common-base and commonemitter large-signal modulation performance are very similar in low bit rates (2.5 Gbps and 10 Gbps). As the bit rate increase the opening of the eyediagram in common-emitter configuration reduces more than the commonbase configuration (20 Gbps and 40 Gbps). For an NRZ signal, there are only two levels: the high level and the low level. The eye height is the difference of the two 3σ points of high and low levels. The 3σ point is defined as the point that is three standard deviations away from the mean value of a probability distribution function (PDF) of each level [66]. From Figure 4.5, for the case of 40 Gbps modulation, common-base has an eye height of 1.6  120  2  2      1  Amplitude (AU)  Amplitude (AU))  4.2. Large-Signal Modulation  3σ   Eye height  0 í  3σ  í í 0  0.02  0.04 0.06 t (ns)  1  3σ   0  Eye height  í 3σ  í 0.08  í 0  0.1  0.02  2  2  1.5  1.5  1  3σ  0.5 Eye height  0 í  3σ  í í 0  1  0.02 0.03 t (ns)  (c)  0.1  0.04  0.05  3σ  0.5 0  Eye height  í 3σ  í 0.01  0.08  (b)  Amplitude (AU)  Amplitude (AU))  (a)  0.04 0.06 t (ns)  0.04  0.05  í 0  0.01  0.02 0.03 t (ns)  (d)  Figure 4.5: Eye-diagram of the digital modulation of the TL in the commonemitter and common-base configurations for different bit-rates. High level and low level are shown with red lines. The eye height is shown as the difference of the two 3σ points of high and low levels. a) 20 Gbps, commonemitter. b) 20 Gbps, common-base. c) 40 Gbps, common-emitter. d) 40 Gbps, common-base. The laser has threshold of IB,th = 1 mA. The amplitude of the data stream is varying from IB,min = 2 mA to IB,max = 4 mA and ER = 5 dB. Parameters of Table 3.1 are used in the simulations.  121  4.2. Large-Signal Modulation AU while common-emitter has an eye height of 1 AU. The enhanced performance of the common-base configuration of the TL is a result of higher  2  2      1  1  Amplitude (AU)  Amplitude (AU))  bandwidth and lower carrier lifetime.   0  Eye height  í    í  í  í  í 0  í 0  0.02  0.04 0.06 t (ns)  0.08  0.1  Eye height  0  0.02  2  2  1.5  1.5  1 0.5 0  Eye height  í í í 0  0.08  0.1  0.04  0.05  (b)  Amplitude (AU)  Amplitude (AU))  (a)  0.04 0.06 t (ns)  1 0.5 0  Eye height  í í  0.01  0.02 0.03 t (ns)  (c)  0.04  0.05  í 0  0.01  0.02 0.03 t (ns)  (d)  Figure 4.6: Eye-diagram of the digital modulation of the TL in the commonemitter and common-base configurations for different bit-rates. High level and low level are shown with red lines. The eye height is shown as the difference of the two 3σ points of high and low levels. a) 20 Gbps, commonemitter. b) 20 Gbps, common-base. c) 40 Gbps, common-emitter. d) 40 Gbps, common-base. The laser has threshold of IB,th = 1 mA. The amplitude of the data stream is varying from IB,min = 2 mA to IB,max = 8 mA and ER = 8.6 dB. Parameters of Table 3.1 are used in the simulations.  For comparison, the performance of the TL in the common-emitter and common-base configurations in the digital modulation (20 and 40 Gbps )  122  4.3. FM Analysis under higher extinction ratio is shown in Figure 4.6. The common-base configuration has a larger eye height for high bit rates than the commonemitter configuration.  4.3  FM Analysis  In this section we compare the large-signal FM performance of the commonemitter and common-base configurations of the TL. For an SCH laser it can be shown that the large-signal FM can be written as [36]:  Δν =  α 4π  1 dP (t) + κP (t) P (t) dt  ,  (4.8)  where α is the linewidth enhancement factor, κ = 2Γ /Vactive hν, and P (t) is the output optical power. We use equation (4.8) to analyze the large-signal FM of the TL. To justify this, we note that from our model in Chapter 3 and Figure 3.3, the variation of the base excess minority carriers is the range of 1016 cm−3 while the QW concentration is in the range of 1018 cm−3 , hence we can attribute the FM to QW carriers and use equation (4.8). α is defined as the variation of the real part of the active region index of refraction (n) due to carrier density variation (N ):  α=−  4π dn , λG0 dN  (4.9)  where G0 is the differential gain and λ is the optical wavelength. The first term in the equation (4.8) is the transient frequency chirping and the second 123  4.4. Discussion term is the adiabatic frequency chirping which gives rise to frequency offset between the on and off levels due to differences in the steady-state values of N. Figure 4.7 shows the FM analysis of the TL in the common-emitter and common-base configurations. Both configurations show the same performance in terms of the adiabatic and the transient frequency chirping in low bit-rate digital modulation. In high bit-rate digital modulation, commonbase performance degrades slightly, since the common-base has lower risetime and fall-time hence should have a higher transient frequency chirping (transient frequency chirping is proportional to the time derivative of the optical power).  4.4  Discussion  In this chapter we studied the large-signal performance of the TL by solving the model introduced in the Chapter 3. TL in the common-base configuration due to enhancement in the small-signal bandwidth shows better large-signal performance, i.e., smaller turn-on delay, capability of high bit rate (> 40 Gbps) digital modulation, bigger opening in eye-diagram. TL in the common-emitter configuration has similar performance as a conventional laser. FM analysis shows that TL does not improve the frequency chirping characteristics.  124  4.4. Discussion  10  8  8  6  6  Δν (GHz)  Δν (GHz)  10  4 2  4  2  0 10  12  14  16  18  0 10  20  t (ns)  12  14  25  25  20  20  15  15  10  10  5 0  0 −5  −10  −10  0.8  0.9 1 t (ns)  (c)  20  5  −5  0.7  18  (b)  Δν (GHz)  Δν (GHz)  (a)  −15  16 t (ns)  1.1  1.2  −15  0.7  0.8  0.9 1 t (ns)  1.1  1.2  (d)  Figure 4.7: Large-signal FM analysis of the TL in the a) 2.5 Gbps, commonemitter configuration and b) 2.5 Gbps, common-base configuration. c) 40 Gbps, common-emitter. d) 40 Gbps, common-base.  125  Chapter 5  Conclusion and Future Works 5.1  Summary and Conclusion  This thesis discussed the dynamics of the conventional SCH laser and TL. Direct modulation of the conventional laser and TL was studied. The effect of the gain compression and carrier transport effect were studied. For this purpose two-level and three-level rate equations were used and modeled and it was found that the effect of the gain compression on small-signal modulation response is different from the carrier transport effect. Integration of the laser rate equations and continuity equation was accomplished by using the quantum capture and escape. A model to predict the large-signal and small-signal modulation of both conventional laser and TL was developed. Quantum mechanical capture and escape were modeled by their lifetimes and reservoir models. However, they can be calculated based on the information obtained through analyzing the QW structure. Such a modeling is discussed in [67], where analytical formulas were derived for the quantum capture and escape lifetimes and they matched with the  126  5.1. Summary and Conclusion experimental results. These fast processes (∼ ps), when considered together, can change the small-signal response by increasing the effective capture lifetime and introducing parasitic effect. Based on the developed model the small-signal modulation response of the conventional VCSEL was modeled. The parasitics effect on the smallsignal modulation was described by a single pole low-pass transfer function and it was shown that the intrinsic response of the SCH laser can be approximated by a third order transfer function. To verify the model, we fabricated and prepared 850 nm VCSELs and performed DC and AC measurements on them. We showed that the model can fit experimental results. In our study, it was shown that the TL has better modulation characteristics in comparison to a conventional laser due to enhanced carrier dynamics. To prove this, the model was extended with the use of proper boundary conditions to include the transistor effect. It was found that common-emitter configuration has the same small-signal modulation response as the SCH laser. However, common-base configuration of the TL has a very different response and its bandwidth can be much higher than the SCH laser, albeit with reduced RF gain. This finding was explained by the reduced lifetime of the carriers in the base region. It was also shown that DC characteristics of the TL are very different depending whether the bias point of the transistor is below or above threshold current of the laser. The DC and AC current gains of the transistor were reduced upon reaching the threshold. The frequency response of the transistor, β(ω), was changed as the resonance of the small-signal modulation of the laser was coupled to the collector current showing the resonance 127  5.1. Summary and Conclusion behavior in the current gain. It was shown that the small-signal modulation response of the TL in both the common-base and common-emitter configurations can be approximated by a third order polynomial transfer function. With this method it was shown that the relaxation octillion frequency is effectively enhanced in the common-base configuration. The effect of feedback on the common-emitter configuration of the TL was studied and it was shown that with feedback we can get a response similar to the common-base configuration. In the large-signal modulation regime the common-base displayed better performance then the commonemitter in terms of the smaller turn-on delay and better eye-diagram opening in higher data bit-rates. FM analysis showed that both configurations have almost the same performance. The limitations of the model can be described in a few directions. Our developed model is based on the diffusion in the base of the BJT, it ignores drift of the carriers. It ignores the thin base effects, band discontinuities, e.g., in an HBT. It is for single QW and the QW is treated as a point in the diffusion equation. Quantum mechanical capture and escape effects are described by their lifetimes. Comparing to the model in [28], our model is more detailed in the sense that it can be used to calculate the carrier distribution, different configurations of TL can be modeled, and more physical insights can be gained about device operation. The model in [68] is very similar to our work.  128  5.2. Future Work  5.2  Future Work  One important task that could be done is to derive the equivalent SPICE circuit from the TL model. Such equivalent circuits have been considered before this study for the QW lasers [43, 69]. For TL we would need to include the equivalent circuit for the transistor. In simple BJT the equivalent smallsignal circuit comes from the junction law and base charge control equation [22] in which the ib is related to the vbe through a first order differential equation: ib = Gse vbe + Cse  dvbe , dt  (5.1)  e e IB and Cse = IB τB which are the transistor AC conkT kT ductance and capacitance, respectively and they are associated with the  where Gse =  emitter-base junction due to charge storage effects. With this approach we can achieve the famous small-signal hybrid-π model of the BJT. However, with the addition of the QW to the base of the transistor we need to change the basic charge control equation to account for the current of spontaneous and stimulated recombination in the QW. This can be done by adding a variable current source and necessary capacitors where their values will depend on the QW characteristics. A similar approach was used in [70]. We may consider using a more elaborate model for the transistor section. In short base BJTs, the boundary conditions described by equations (3.12) and (3.56) need to be updated [71]. It has been shown that the following boundary conditions can model the short base effect accurately [64]:  δn(0) =  JE + n∗E , 2evR  (5.2) 129  5.2. Future Work  δn(WB ) = where vR =  −JC , 2evR  (5.3)  kB T /2πm is a unidirectional velocity associated with the  Maxwellian velocity distribution. In HBTs, there may be a band spike in the emitter base junction. Band discontinuity causes thermionic emission and quantum mechanical tunneling which can distort the velocity distribution of the injected electrons from Maxwellian form. A simple charge density continuity equation cannot be used in this circumstance. By solving the Boltzmann transport equation, it is shown that in the presence of the band discontinuity drift-diffusion model may fail [72]. Another important aspect of the model would be comparing the model predictions with the experimental results. To the best of our knowledge there has been just one published measurement result for the direct modulation of TL in [26]. In that work, the authors reported a 13.5 GHz small-signal bandwidth for the common-emitter configuration of an edge-emitting laser. Our group is working on the TVCSELs and as soon as measurement results are obtained we will be able to compare the model predictions with actual experimental data.  130  Bibliography [1] A. Yariv and P. Yeh, Photonics: Optical electronics in Modern Communications. OXFORD, 2007. [2] T. H. Maiman, “Stimulated optical radiation in ruby,” Nature, vol. 187, pp. 493 – 494, 1960. [3] R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, and R. O. Carlson, “Coherent light emission from GaAs junctions,” Phys. Rev. Lett., vol. 9, pp. 366 – 368, 1962. [4] N. Holonyak, Jr., and S. F. Bevacqua, “Coherent (visible) light emission from GaAs(1−x) Px junction,” Appl. Phys. Lett., vol. 1, pp. 82 – 83, 1962. [5] M. I. Nathan, W. P. Dumke, G. 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Zah, “Bandwidth enhancement with tunable optical equalizer for high speed intensity modulation,” in Lasers and Electro-Optics, 2008 and 2008 Conference on Quantum Electronics and Laser Science. CLEO/QELS 2008. Conference on, pp. 1–2, 2008. [64] O. Hansen, “Diffusion in a short base,” Solid State Electronics, vol. 37, pp. 1663 – 1669, 1994. [65] M. S. Demokan and A. Nacaroglu, “An analysis of Gain-Switched Semiconductor Lasers Generating Pulse-Code-Modulated Light with a High Bit Rate,” IEEE Journal of Quantum Electronics, vol. QE-20, pp. 1016 – 1022, 1984. [66] MATLAB R2009a online help. [67] B. Romero, J. Arias, I. Esquivias, and M. Cadab, “Simple model for calculating the ratio of the carrier capture and escape times in quantumwell lasers,” Appl. Phys. Lett., vol. 76, pp. 1504 – 1506, 2000. [68] M. Shirao, S. Lee, N. Nishiyama, and S. Arai, “Large-signal analysis of a transistor laser,” Quantum Electronics, IEEE Journal of, vol. 47, pp. 359 –367, march 2011. [69] S. C. Kan and K. Y. Lau, “Intrinsic equivalent circuit of quantum-well lasers,” IEEE Photon. Technol. Lett., vol. 4, pp. 528 – 530, 1992. [70] H. W. Then, M. Feng, and H. N., “Microwave circuit model of the threeport transistor laser,” Journal of Appl. Phys., vol. 107, pp. 094509–1 – 094509–7, 2010. [71] D. L. Pulfrey, A. R. S. Denis, and M. Vaidyanathan, “Compact Modeling of high-frequency, small-dimension bipolar transistor,” IEEE Proceedings on Optoelectronic and Microelectronic Materials Devices, pp. 81 – 85, 1999. [72] A. R. S. Denis and D. L. Pulfrey, “Quasiballistic transport in GaAsbased heterojunction and homojunction bipolar transistors,” Journal of Applied Physics, vol. 84, pp. 4959 – 4965, 1998.  137  Appendix A  Derivation of the Kirchhoff ’s Current Law in the Transistor Laser For the transistor laser with 3-ports for the electrical signals and one optical port, equation (3.15) can be proven by applying the continuity equation to the electrons and holes in the base region. We start from the general form of continuity equation: 1 δn ∂δn + ∇ · je = , q τB ∂t  (A.1)  where je is the electron current density in the base region, τB is the recombination lifetime of the carriers, and δn is the carrier distribution. We can convert this differential equation to an integral equation by integrating the equation A.1 in the closed volume of base region:  (∇ · je ) dv = V  Qn dQn + . τB dt  (A.2)  138  Appendix A. Derivation of the Kirchhoff’s Current Law in the Transistor Laser In equation (A.2), V is the base region volume, Qn is the total electrons charge in the base. By using the Gauss theorem, the left hand side of the equation (A.2) may be replaced by surface integral of the electron current density which equals the total current of the electrons in the base:  (∇ · je ) dv = V  je · dS = Δie ,  (A.3)  S  where S denotes the surface that encloses the base volume, and Δie is the total current of the electrons that flows into base region. Combining equations (A.2) and (A.3) leads to:  Δie =  Qn dQn + , τB dt  (A.4)  which is the equation for the charge control analysis that can be used, e.g., in pulse analysis of the BJTs. A very similar equation holds for the holes since the base is assumed to be neutral:  − Δih =  Qn dQn + , τB dt  (A.5)  where Δih is total current of the holes that flows into base region. Combining equations (A.4) and (A.5) leads to Krichhof’s current law in the TL:  Δin + Δih = 0 .  (A.6)  139  Appendix B  Derivation of Charge Conservation for the Virtual States in the QW Region Equation (3.19) can be proven by applying the continuity equation to the virtual states in the QW region. Decomposing the charge in the QW to nV.S. and nQW , we can write the continuity equation for the virtual states carriers: 1 δnV.S. ∂δnV.S. + ∇ · je = , q τS ∂t  (B.1)  where we have shown the recombination lifetime by τS . Repeating the same steps as in Appendix A and noting that there are two charge flow mechanisms, i.e., diffusion current (jV.S. ) and QW current (jQW ), the divergence term can be written as:  (∇ · je ) dv = V  je · dS = iV.S. − iQW .  (B.2)  S  Combining the equations (B.1) and (B.2), we obtain equation (3.19).  140  Appendix C  Publications C.1  Peer-Reviewed Journal Publications  W. Shi, B. Faraji, M. Greenberg, J. Berggren, Y. Xiang, M. Hammar, M. Lestrade, Z. Li, S. Li, L. Chrostowski, “Design and modeling of a transistor vertical-cavity surface-emitting laser,” Optical and Quantum Electronics, 1 - 8, 2011 (Invited) B. Faraji, W. Shi, D. L. Pulfrey, L. Chrostowski, “Analytical modeling of the transistor laser,” IEEE Journal of Selected Topics in Quantum Electronics, 15(3), 594 - 603, 2009 B. Faraji, W. Shi, D. L. Pulfrey, L. Chrostowski, “Common-emitter and common-base small-signal operation of the transistor laser,” Applied Physics Letters, 93(14), 2008. B. Faraji, D. L. Pulfrey, L. Chrostowski, “Small-signal modeling of the transistor laser including the quantum capture and escape lifetimes,” Applied Physics Letters, 93(10), 2008. W. Shi, L. Chrostowski, B. Faraji, “Optical saturation and voltage control of a transistor vertical cavity surface emitting laser,” IEEE Photonics Technology Letters, 20(24), 2141 - 2143, 2008. L. Chrostowski, B. Faraji, W. Hofmann, M.-C. Amann, S. Wieczorek, W. W. Chow, “40 GHz bandwidth and 64 GHz resonance frequency in injectionlocked 1.55 μm VCSELs,” IEEE Journal of Selected Topics in Quantum Electronics, 13(5), 1200 - 1208, 2007.  141  C.2. Peer-Reviewed Conference Papers E. Bisaillon, D. T. H. Tan, B. Faraji, A. G. Kirk, L. Chrostowski, D. V. Plant, “High reflectivity air-bridge sub-wavelength grating reflector and Fabry-Perot cavity in AlGaAs/GaAs,” Optics Express, 14(7), 2573 - 2582, 2006. A. K. M. Lam, L. Chrostowski, B. Faraji, R. Kisch, N. A. F. Jaeger, “Modified optical heterodyne, down-conversion system for measuring frequency response of wide-band wavelength-sensitive electro-optical devices,” IEEE Photonics Technology Letters, 18(20), 2183 - 2185, 2006.  C.2  Peer-Reviewed Conference Papers  B. Faraji, N. A. F. Jaeger, L. Chrostowski, “Modelling the effect of the feedback on the small signal modulation of the transistor laser,” Photonics Society Annual Meeting, Denver, Colorado, US, 2010. X. Wang, B. Faraji, L. Chrostowski, “Interference effects on the frequency response of injection-locked VCSELs,” IEEE International Semiconductor Laser Conference, Kyoto, Japan, 2010. W. Shi, B. Faraji, and L. Chrostowski, “Self-consistent modeling of a transistor vertical-cavity surface-emitting laser,” 10th International Conference on Numerical Simulation of Optoelectronic Devices, Atlanta, US, 2010. (invited) B. Faraji, W. Shi, D. L. Pulfrey, L. Chrostowski, “Small-signal modeling of the transistor laser in common-emitter and common-base configurations,” IEEE Lasers and Electro-Optics Society Conference, Newport Beach, US, 2008. W. Shi, L. Chrostowski, B. Faraji, “Voltage controlled operation of a transistor vertical cavity surface emitting laser,” IEEE International Semiconductor Laser Conference, Sorrento, Italy, 2008. Q. Gu, K. W. Chan, A. Au, B. Faraji, L. Chrostowski, W. Hofmann, M.C. Amann, “Optical injection locking of VCSEL with amplitude modulated master,” IEEE International Topical Meeting on Microwave Photonics, pp. 183 - 186, 2007. 142  C.2. Peer-Reviewed Conference Papers  B. Faraji, L. Chrostowski, W. Hofmann, M.-C. Amann, “Dual-resonance frequency response in injection-locked 1.55 μm VCSELs,” Optical Fiber Communication Conference, Anaheim, US, March, 2007. L. Chrostowski, B. Faraji, W. Hofmann, R. Shau, M. Ortsiefer, M.-C. Amann, “40 GHz bandwidth and 64 GHz resonance frequency in injectionlocked 1.55 μm VCSELs,” IEEE International Semiconductor Laser Conference, Hawaii, US, 2006. E. Bisaillon, D. T. H. Tan, B. Faraji, Y. Zeng, C. Ostafew, R. KrishnaPrasad, L. Chrostowski, D. V. Plant, “Resonant grating based Fabry-Perot cavity in AlGaAs/GaAs,” IEEE Lasers and Electro-Optics Society Conference, Montreal, Canada, 2006. D. T. H. Tan, B. Faraji, N. Zangenberg, T. Tiedje, L. Chrostowski, E. Bisaillon, D. V. Plant, “A novel method for fabrication of free standing subwavelength grating is GaAs/AlGaAs,” Canadian Conference on Electrical and Computer Engineering, Ottawa, Canada, 2006. B. Faraji, E. Bisaillon, D. T. H. Tan, D. Plant, L. Chrostowski, “Finitesize resonant sub-wavelength grating high reflectivity mirror,” IEEE Lasers and Electro-Optics Society Conference, Montreal, Canada, 2006.  143  

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