Research Collection
Doctoral Thesis
Modeling the optical processes in semiconductor lasers
Author(s): Witzig, Andreas
Publication Date: 2002
Permanent Link: https://doi.org/10.3929/ethz-a-004407405
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ETH Library Diss. ETH No. 14694
Modeling the Optical Processes in Semiconductor Lasers
A dissertation submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH
for the degree of Doctor of Technical Science
presented by
ANDREAS WITZIG Dipl. El.-Ing. ETH born 26 10 1973 citizen of Laufen-Uhwiesen, Switzerland
accepted on the recommendation of
Prof. Dr. Wolfgang Fichtner, examiner Prof. Dr. Hans Melchior, co-examiner
2002 Contents
Acknowledgements vii
Abstract ix
Zusammenfassung xi
1 Introduction 1 1.1 Motivation 1 1.2 Active Optoelectronic Devices 2 1.3 Optoelectronic Device Simulation 6 1.4 Contents 7
2 Physics of Optoelectronic Devices 9 2.1 Introduction 9 2.2 Basic Semiconductor Laser Physics 10 2.3 Empirical Laser Model: The Lumped Laser Equations 11 2.4 Eigenmode Laser Equations 18 2.4.1 Electro-Thermal Model 18 2.4.2 Optical Model 19 2.4.3 Local Electron-Photon Interaction 23 2.4.4 Device-Specific Models 26
2.4.5 Extension of the Eigenmode Laser Equations . 28 2.4.6 Beyond the Adiabatic Approximation 30 2.5 More Fundamental Laser Models 31 2.5.1 Semiconductor Bloch Equations 32
2.5.2 Slowly Varying Amplitude Approximation ... 32
i ii CONTENTS
2.6 Characteristic Time Scales 33 2.7 Diffraction Theory 36 2.8 Discussion 39
3 Optical Cavities and Waveguides 41 3.1 Introduction 41 3.2 Optical Cavities 42 3.2.1 Empirical Cavity Model 42 3.2.2 Cavity Eigenvalue Equation 44 3.2.3 Mode Designation for VCSELs 46 3.2.4 ID Cavity 49 3.2.5 Spectral Properties 52 3.3 Optical Waveguides 54 3.3.1 Waveguide Eigenvalue Equation 54 3.3.2 Slab Waveguides 55 3.3.3 Waveguide Spectrum 56 3.4 Mathematical Background 59 3.4.1 Eigenvalues and Eigenvectors 59 3.4.2 Quasi-Normal Modes 60 3.5 Discussion 62
4 Implementation 65 4.1 Introduction 65 4.2 Analysis of the Governing Equations 66 4.3 Traveling-Wave Optics 67
4.3.1 Finite-Difference Time-Domain Method .... 67 4.3.2 Transfer-Matrix Method 68 4.4 Hierarchy of Approximations for Waveguide Solutions 70 4.5 Numerical Eigenmode Solution 73 4.6 Eigenmode Laser Equations 74 4.7 Discussion 78
5 Results 81 5.1 Introduction 81 5.2 Edge-Emitting Laser Diodes 82 5.2.1 Radiation Leakage into Substrate 82 5.2.2 Spontaneous Emission Coupling 83 CONTENTS m
5.3 Surface-Emitting Lasers 89
5.3.1 Finite-Element Optical Mode Calculation ... 89 5.3.2 Optical Modes Derived by FDTD 90 5.4 Discussion 95
6 Conclusion and Outlook 99 6.1 Major Results 99 6.2 Further Development 100
A Maxwell's Equations 105
B Semiconductor Dispersion Relations 107
C Coupled Semiconductor Equations 111
D Notation 115
Bibliography 120
Curriculum Vitae 131
List of Figures
1.1 Edge-Emitting Laser 4 1.2 Vertical-Cavity Surface-Emitting Laser 5 1.3 Semiconductor Laser Design Methodology 6
2.1 Laser Characteristics 13 2.2 Modulation Response 14 2.3 Time-Domain Response 15
2.4 3D Optical Field in a DFB Laser 29 2.5 Schematic Device Physics for Diode Lasers 35
2.6 Geometry Definition for the Diffraction Theory .... 36
3.1 Body-of-Revolution Expansion of VCSEL Modes ... 48 3.2 Optical Eigenmode of a ID Cavity 50 3.3 Spectral Properties of Bragg Mirrors 52 3.4 Slab Waveguide 55 3.5 Waveguide Spectrum of Slab Structure 56
4.1 Simple Simulation Flow 76 4.2 Simulation Flow for Optical Re-Solving 76
5.1 Radiation Leakage into Substrate 82 5.2 Results of the Waveguide Eigenvalue Problem 84 5.3 Setup and Results of the FDTD Simulation 85 5.4 EEL-Simulation: Time-Domain Results 86
5.5 EEL-Simulation: Relative Power vs. Distance 87 5.6 Finite-Element Result of VCSEL Eigenmodes 90 5.7 FDTD Result of Steady-State VCSEL Field 91
v VI LIST OF FIGURES
5.8 Mode Pattern in the Active Region 92 5.9 Comparing Devices with Different Aperture Diameters 93
5.10 3D Simulation of a Tilted-Pillar VCSEL 94 Acknowledgements
First of all, I would like to thank Prof. Dr. Wolfgang Fichtner for his support and for his confidence in my work. He provided an excellent working environment and put a special effort into our group. Only through him, we got to establish the connections to leading research groups and industrial partners which proved to be essential for our work. I also wish to thank Prof. Dr. Hans Melchior for accepting to be the co-examiner of this thesis and for carefully reviewing the manuscript. My interest in numerical simulations was initially sparked in an undergraduate student project. I am grateful to Dr. Pascal Leucht¬ mann who supported this project actively and taught me the secrets of electromagnetics. Furthermore, one of the reasons I finally ended up in optoelectronics was an ERASMUS exchange project with Bristol University, UK. I would like to thank Prof. Dr. John Gowar for giving me this opportunity and for spending a lot of time in discussions with me. My start at the Integrated Systems Laboratory (Institut für Inte¬ grierte Systeme, IIS) has been in the group of Dr. Peter Regli from whom I learned a lot about programming and software integration. The close collaboration with Dr. Christian Schuster, Dr. Thomas Körner, and Andreas Christ was fruitful and both a personal and professional enrichment. At this time, Dr. Bernd Witzigmann started with the laser extension of the device simulator DESSIS-ISE. I ad¬ mire Bernd for many decisions that he made in the early stage of this project. I wish to thank him for his patience and for explaining over and over again the details of semiconductor physics. Michael Pfeiffer, Matthias Streiff, Lutz Schneider, and Serguei
Vll vm Acknowledgements
Chevtchenko joined the group during the last three years. The in¬ teraction within our group has become considerable, and the Opto¬ electronics Modeling Group gained momentum. A great part of the motivation in my work I had from the good atmosphere in our office.
I thank them all very much! As the Optoelectronics Modeling Group expanded, administrative issues and funding got some importance. I wish to express my grat¬ itude to Dr. Dölf Aemmer, the coordinator of the research activities in technology computer aided design (TCAD) at IIS. He was always there for us and I learned a lot from him about the formulation and planning of new projects. From the beginning of the optoelectronics activities at IIS, the col¬ laboration with industry was very important. First of all comes the ISE Integrated Systems Engineering AG who provided the basis for a professional software development. I am grateful to the ISE soft¬ ware developers who showed a lot of confidence in our work. On the hardware side, we had the chance to collaborate with a laser manufac¬ turer that was an off-spring of the IBM Research Lab in Riischlikon, later owned by JDS Uniphase and then by Nortel Networks. The
collaboration with the strong R&D group was very profitable. Later, collaborations with Avalon Photonics Inc. and Opto Speed AG have been established. I want to thank all these partners for their active support.
During my work at IIS I had the opportunity to advise or sev¬ eral students performing undergraduate research in our group. There
has been considerable output from their work and we had many in¬ teresting discussions. I wish to thank Volker Mornhinweg, Thomas Eberle, Pamela Lässer, Kersten Schmidt and Adrian Bregy. In regard of the student projects, I must also thank Dr. Norbert Felber, who coordinated them.
In an advanced stage of my work, Prof. Dr. Joachim Piprek invited me to the University of California in Santa Barbara (UCSB). The work at UCSB was funded by a joint ETH-UCSB scholarship that supports research collaborations between the two institutions. It was an important experience for myself and an invaluable enrichment for the development of our simulator. Abstract
In modern telecommunication systems, semiconductor lasers are key components. This doctoral thesis presents a physics-based approach to simulate active optoelectronic devices. It covers the analysis of a broad range of devices, including Edge-Emitting Lasers (EELs) and Vertical-Cavity Surface-Emitting Lasers (VCSELs). A numeri¬ cal solver for the optical field has been implemented, and an interface to an electro-thermal device simulator has been built. Contributions have been made in the formulation of the physical models, and in the coupling scheme between electronics and optics. For EELs, the optical field can be separated into transverse waveg¬ uide modes and longitudinal cavity modes. The transverse eigen¬ modes have been solved in 2D applying a general but efficient finite- element method. The longitudinal modes have been calculated using the transfer-matrix method. For VCSELs, a separation into trans¬ verse and longitudinal direction is not possible. Instead, the fields can be expanded into a Fourier series and the eigenmodes are solved in 2D. In addition to the finite-element solution of the optical eigenvalue problem, the finite-difference time-domain method has been applied for the calculation of the electromagnetic wave propagation. In a post¬ processing step, the optical eigenmodes of general VCSELs have been obtained. Furthermore, the spontaneous emission coupling factor is calculated rigorously by a full-3D treatment of an EEL structure. The work presented here has to be seen in the context of the activi¬ ties of the Optoelectronics Modeling Group at the Integrated Systems Laboratory (IIS) of the Swiss Federal Institute of Technology (ETH) in Zurich, Switzerland. In a joint effort, this group is building a com¬
ix X Abstract
prehensive physics-based device simulation tool for optoelectronics. Optoelectronic device simulation is a scientific challenge. Numer¬ ous questions need to be answered both on the theory of numerical modeling and on the applied device physics. In addition, two as¬ pects have been important for the successful implementation of the laser simulator: First, a close collaboration with device manufacturers has allowed to share detailed device specifications as well as measure¬ ment results. With this data, the simulator could be validated and calibrated. The industrial device designers, in return, have got ac¬ cess to the simulator at an early stage of the software development. Second, a professional software environment is essential for both the development and the application of the laser models. The simulator developed in the Optoelectronics Modeling Group is built as an ex¬ tension to the general device and circuit simulator DESSIS-ISE. The work of many PhD theses has provided the scientific basis of this sim¬ ulator. Since the spin-off of ISE Integrated Systems Engineering AG from IIS in 1993, the computer program has been commercialized. In
consequence, the new laser simulator is readily equipped with state-of- the-art tools for geometry editing, mesh generation, and visualization. In numerous discussions with project partners and users of device simulation software, the gap between numerics and industrial device design became evident. This work bridges theory and application. It provides an effective means for the numerical calculation of the optical fields in laser cavities and incorporates them into a versatile device
simulator. The laser extension of DESSIS-ISE has been attested a high utility for the design of advanced optoelectronic devices. Zusammenfassung
In der modernen Telekommunikation sind Halbleiterlaser Schlüssel¬ komponenten. Die vorliegende Doktorarbeit befasst sich mit der ri¬ gorosen Simulation dieser aktiven optoelektronischen Bauelemente. Dabei werden kantenemittierende Laser ('edge-emitting lasers') sowie oberflächenemittierende Laser mit vertikalen Resonatoren ('vertical- cavity surface-emitting lasers') abgedeckt. Es wurde eine numeri¬ sche Lösung für das optische Feld implementiert und ein Interface erstellt zwischen dem Optik-Löser und einem bestehenden elektro- thermischen Simulator. Die Beiträge zur Kopplung zwischen Optik und Elektronik bestehen in der Formulierung der physikalischen Mo¬ delle sowie im Kopplungsschema und der praktischen Implementie¬ rung. Das optische Feld in kantenemittierenden Lasern kann in trans¬ versale und longitudinale Moden separiert werden. Die transversalen Moden wurden mit der Methode der Finiten Elemente in 2D gelöst. Für die longitudinalen Moden wurde die Transfer-Matrix-Methode verwendet. Für oberflächenemittierende Laser ist eine Separation in transversale und longitudinale Richtung nicht möglich. Die Felder können jedoch in eine Fourier-Reihe entwickelt werden, und die Ei¬ genmoden werden in 2D gelöst. Zusätzlich zur Lösung der Eigenmoden mit der Methode der Fi¬ niten Elemente wurde die Wellenausbreitung mit Finiten Differenzen im Zeitbereich ('finite-difference time-domain') berechnet. Aus die¬ sen Resultaten konnten die optischen Moden von allgemeinen ober¬ flächenemittierenden Lasern bestimmt werden. Ausserdem konnte mit einer voll-vektoriellen 3D-Rechnung die Kopplung der spontanen Emission in kantenemittierenden Lasern rigoros bestimmt werden.
XI Xll Zusammenfassung
Die vorliegende Arbeit muss im Zusammenhang gesehen werden mit der Tätigkeit der 'Optoelectronics Modeling'-Gruppe des Insti¬ tuts für Integrierte Systeme (IIS) an der Eidgenössischen Technischen Hochschule Zürich. Gemeinsam wird daran gearbeitet, ein Simula¬ tions-Werkzeug für die Optoelektronik bereitzustellen, das beim in¬ dustriellen Bauelement-Entwurf nützlich ist. Optoelektronik-Bauteilsimulation ist eine wissenschaftliche Her¬ ausforderung, welche Fragestellungen aus der numerischen Model¬ lierung und der Physik der Bauteile einschliesst. Für die erfolgrei¬ che Implementierung eines Bauteil-Simulators sind zwei zusätzliche Aspekte erforderlich: Erstens erlaubte die enge Zusammenarbeit mit Bauelement-Herstellern den Zugang zu detaillierten Bauteil-Spezifika¬ tionen und zu Messdaten. Damit konnten die Simulationen überprüft und Unsicherheiten in den Materialparametern eliminiert werden. Als Gegenleistung haben die Bauteil-Entwickler den Simulator schon in ei¬ nem frühen Stadium zur Lösung ihrer Probleme verwenden können. Zweitens ist eine professionelle Software-Umgebung essentiell für die Entwicklung und Anwendung der Software. Der Laser-Simulator wur¬ de als Erweiterung des Bauteil- und Schaltungs-Simulators DESSIS- ISE entwickelt. Die wissenschaftliche Basis dieses Simulators wurde in etlichen Doktorarbeiten am IIS geschaffen. Mit der Gründung der Fir¬ ma ISE Integrated Systems Engineering AG, einer Spin-Off des IIS, wurde das Programm komerzialisiert. Da sie Laser-Modelle direkt in DESSIS-ISE eingebaut wurden, standen von Anfang an moderne Software-Werkzeuge für das Erstellen der Geometrie, die Erzeugung eines Gitters sowie für die Visualisierung zur Verfügung. In den vielen Diskussionen mit den Projektpartnern und Anwender von Bauteil-Simulatoren wurde klar, dass immer noch ein Graben be¬ steht zwischen den numerischen Problemen einerseits und dem indu¬ striellen Bauteil-Design andererseits. Die vorliegende Arbeit soll eine Brücke schlagen zwischen Theorie und Praxis. Es wurde ein exakter und effizienter Löser für die optischen Felder in Laser-Resonatoren er¬ stellt und in einen vielseitigen Bauelement-Simulator eingebaut. Die Laser-Erweiterung von DESSIS-ISE wurde bereits erfolgreich für das Design von hochentwickelten Optoelektronischen Komponenten ein¬ gesetzt und von den Anwendern als nützliches Werkzeug eingestuft. Chapter 1
Introduction
1.1 Motivation
Active optoelectronic devices have become important components in the modern world. Semiconductor lasers have matured rapidly and have become reliable and relatively low-cost. The device physics as well as the design parameters on the system level are fairly well under¬ stood [1]. Competition between manufacturers drives optimization of existing devices and development of novel device concepts. Processing has changed from a laboratory environment to mass fabrication. As a result, physical modeling has become an integral part of industrial en¬ gineering and development. The demand for a simulation software for optoelectronic devices motivates the research presented in this thesis and ongoing projects in this area.
A similar transition in industrial device design has evolved more than a decade ago with microelectronic devices based on silicon. Along with the change in the development focus, technology computer aided design (TCAD) was introduced and in the mean time has become indispensable [2]. Device modeling has matured especially with regard to the numerical solution of the strongly coupled set of equations that results from a physics-based software approach. Compared to silicon device modeling, it turns out that laser simu¬ lation is more complex due to the additional requirement of an optical
1 2 CHAPTER 1. INTRODUCTION
wave solution. Solving the laser device equations is a challenge both in formulating the physical models as well as in the numerical treatment. With the introduction of quantum well active regions, the semicon¬ ductor equations have to be extended: Schrödinger's equation has to be solved and new transport phenomena have to be considered [3, 4].
In this thesis, the optical processes in semiconductors have been analyzed and put into the context of comprehensive laser simulation. The evaluation and implementation of the physical models have been done with the aim to provide an accurate and robust numerical solver that helps the device designer in the development of optoelectronic devices.
1.2 Active Optoelectronic Devices
This work focuses on semiconductor light sources. In the following overview, the most important active optoelectronic devices are re¬ viewed. Some basic semiconductor physics is anticipated and will be explained in more detail in Sec. 2.2.
The application areas of active optoelectronic devices include op¬ tical communication systems, the sensing and measurement area as well as display and lighting applications. Key characteristics are the efficiency, spectral properties, modulation capabilities, reliability and the possibility for a low-cost fabrication.
Light-Emitting Diodes
Light-Emitting Diodes (LEDs) are semiconductor p-n junctions oper¬ ated in forward-bias. LEDs can be designed for the emission of light in the ultraviolet, visible, or infrared regions of the electromagnetic spec¬ trum. The radiation is incoherent and the spectral spread (linewidth) is between 10 and 50 nm, which is relatively narrow compared to ther¬ mal radiation but broad in comparison to lasers. In optical communi¬
cation applications, the figure of merit of an LED is the optical power that can be coupled into an optical fibre at a given diode current. Common device structures include both the edge-emitting as well as the surface-emitting configuration described below. The modulation 1.2. ACTIVE OPTOELECTRONIC DEVICES 3
bandwidth is limited by the carrier lifetime in the device (maximum modulation frequency ~ 100 MHz) [5]. Optical feedback is introduced in the design of super-luminescent light-emitting diodes (SLEDs) [6, 7] or resonant-cavity light-emitting diodes (RCLEDs) [8]. As a result, the devices have higher slope effi¬ ciency, the modulation bandwidth can be increased to about 3-5 GHz, and the linewidth is 5-10 times narrower than for a similar LED.
Furthermore, the fiber coupling is improved due to a more directed emission pattern, higher intensity, and higher spectral purity.
Edge-Emitting Lasers
Coherent light sources employ a different mechanism for the gener¬ ation of light. Due to the optical feedback, stimulated emission be¬ comes dominant over spontaneous emission (see Sec 2.1). As a result, lasers have a high spectral purity (linewidth ^ 0.1 nm) and a highly directional light beam. High modulation capabilities, good coupling into optical fibers and small device size makes semiconductor lasers the ideal light source for telecomm applications, as for example in the realization of wavelength-division multiplexed (WDM) systems [5]. The traditional laser structure is the Edge-emitting laser (EEL), as shown in Figure 1.1. The light is confined laterally by a transverse optical waveguide and, in longitudinal direction, by partially trans¬ parent mirrors. Referring to the coordinate system in Figure 1.1, the laser output is in the ^-direction. A layer structure of different ma¬ terials confines the light in the ^/-direction. In the x-direction, the waveguide is formed by a ridge structure.
The first fabrication step of a laser is the growth of the layer struc¬ ture. The so-called epitaxy produces excellent material purity and al¬ lows to control the layer structure almost to the atomic length scale. The ridge structure is etched subsequently. A re-growth of additional layers is possible, resulting in a buried heterostructure laser.
For a Fabry-Perot (FP) laser, the longitudinal confinement con¬ sists of two discrete mirrors. Alternatively, distributed feedback (DFB) lasers have a grating typically along the entire z-direction while the Distributed Bragg reflector (DBR) lasers have grating reflectors only in the passive sections [9]. 4 CHAPTER 1. INTRODUCTION
Figure 1.1: Edge-Emitting Laser (EEL): A ridge structure guides the optical wave, and at the same time provides lateral current confine¬ ment. The optical feedback is provided by the reflection at the front and back facets.
Considerable research interest is in the search of an optimal struc¬ ture such as a broad area laser, a taper or a laser array, aiming at a stable, high-power output beam [10, 11, 12]. With a similar vertical and lateral device structure, but anti-reflective coatings at the front and rear mirrors, a single-pass semiconductor optical amplifier (SOA) can be designed [13]. For this device, light is injected at one side of the device and is amplified in the longitudinal direction.
Active research is also in the design of tunable lasers [14], and in the development of photonic integrated circuits [15]. Presently, the most widely used setup for photonic integration is the multi¬ section structure, similar to Figure 1.1, but with several sections in z-direction. 1.2. ACTIVE OPTOELECTRONIC DEVICES 5
Light Output
Contact ^ Top Air Post ^ Top Mirror Active Region
Figure 1.2: Vertical-Cavity Surface-Emitting Laser (VCSEL): The op¬ tical cavity is formed by distributed mirrors in vertical direction. An air-post structure provides a lateral confinement for both the optical field and the current.
Surface-Emitting Lasers
In vertical-cavity surface-emitting lasers (VCSELs) [7], the optical beam propagates in the direction of the epitaxial growth, and the opti¬ cal feedback is achieved by a layered media that serves as a distributed mirror. In a simple design such as the one displayed in Figure 1.2, the so-called Bragg mirrors consist of two semiconductor materials, grown in quarter-wavelength layers in alternating sequence. Due to the low refractive index contrast of the materials available, 25 - 60 layers are needed to achieve the required reflectivity. A Bragg mirror is placed below and above the active region, and mirror reflectivities of > 99 % are required because the overlap between the optical field and the active region is small.
VCSELs have been proposed two decades ago [16]. They have a high potential in modern telecommunication applications due to 6 CHAPTER 1. INTRODUCTION
Idea / Needs Old Device Idea / Needs Old Device J- Lite rature Model Material i^—»|tCAD f~ J Ab Initio Calc. [Design Test Series Database Optimization Processing Old Devices Physical O o Models TCAD Measurements Analysis m Processing i— Comparison with o Choose Best Device Measurements Measurements > I—
Test Structures 01
> TCAD Analysis New Device I Comparison with New Device Measurements
Figure 1.3: Design methodology for optoelectronic device develop¬ ment. Left: Traditional methodology based on experiments. Right: Modern optoelectronics device design using a combined approach us¬ ing both numerical modeling and experiments.
their circular output beam and, as a consequence, their good coupling efficiency into optical fibers. The photon round-trip time is small and parasitics can be kept at a minimum due to the small device dimensions. Therefore, VCSELs are suited for high-speed modula¬ tion. Furthermore, VCSELs can be produced at low cost so that they are also suitable for consumer applications. More exotic designs include circular-grating surface-emitting lasers (CGSEL) [17], whis¬ pering gallery [18] and photonic band-gap lasers [19].
1.3 Optoelectronic Device Simulation
Traditional development of novel device structures has been guided by an experimental approach. Starting from an existing device, pro¬ totypes of an improved structure are fabricated and measurements are carried out to evaluate the device performance. Device designers learn primarily from measurement results and the loop in the design cycle has to be passed several times before a new device is ready for production. While the empirical approach is expensive and time consuming, numerical device simulation speeds up the design cycle [20]. Prior 1.4. CONTENTS 7
to the fabrication of a prototype, a novel device design is analyzed and optimized by the use of a device simulator. As suggested in Fig¬ ure Figure 1.3, the fabrication of test samples and its measurement are not obsolete in the new design methodology, but the number of process runs in the design cycle is reduced. To build up a reliable material database, the models have to be calibrated by the use of test structures.
In a simulation hierarchy, the most basic investigations are on the material-level and include the calculation of carrier mobilities [21] or gain functions [22]. While the material-level models are typically dimensionless, the device-level simulations account for the geomet¬ rical extensions and the interfaces between several materials. The material parameters, including its dependence on variables such as carrier density or lattice temperature, are the basis for device simu¬ lation. The main goal is to predict measurable device characteristics (e. g. current-power curve or modulation response). In addition, inter¬ nal characteristics such as current density or temperature distribution can be studied. On top of the material and the device level are the system-level simulations, where external circuitry can be introduced and the interplay between several optoelectronic devices can be inves¬ tigated. This work focuses on the device-level.
1.4 Contents
This thesis is organized as follows:
Chapter 1: Introduction. Motivation is given for physics-based
simulation of optoelectronic devices and the most important op¬ toelectronic devices are reviewed. The application of predictive modeling in an industrial environment is discussed.
Chapter 2: Physics of Optoelectronic Devices. First, basic semiconductor laser physics is explained. Then, an equivalent circuit model for semiconductor lasers is presented. A hierar¬ chy of approximations for semiconductor laser simulation is dis¬ cussed, with a special focus on the electron-photon interaction.
Chapter 3: Optical Cavities and Waveguides. For the discus¬ sion of the eigenvalue nature of the optical modes, cavities and 8 CHAPTER 1. INTRODUCTION
waveguides are treated separately. For both of them, empir¬ ical formulas are reviewed for reference, an interpretation of the eigenpair is given, and the spectral properties are discussed. Furthermore, the quasi-normal modes are introduced.
Chapter 4: Implementation. The implementation of a compre¬ hensive laser simulator is presented and a brief review of the numerical methods is given.
Chapter 5: Results. Simulation results are presented for edge-emit¬ ting lasers as well as for VCSELs. The examples include radia¬ tion leakage into substrate, spontaneous emission coupling and the calculation of the 3D optical mode pattern in VCSELs.
Chapter 6: Conclusion and Outlook. Conclusions are drawn and possible future development is discussed in an outlook.
Appendix: Semiconductor dispersion characteristics are reviewed and a complete list of equations for semiconductor laser simula¬ tion is presented. Chapter 2
Physics of Optoelectronic Devices
2.1 Introduction
The optimal choice of physical models is a considerable challenge in optoelectronic device simulation. In the trade-off between model ac¬ curacy and computational complexity, it is important to evaluate the physical effects to be included. In the mathematical derivation of the models, several approximations are made. This chapter discusses the physical models suitable for laser simulation.
The basic semiconductor laser physics is recaptured in Sec. 2.2. In order to derive a common notation with the classical laser theory, the zero-dimensional formulation is outlined briefly in Sec. 2.3. With this set of empirical rate equations, the laser characteristics can be repro¬ duced. The laser simulator employs the eigenmode laser equations, as presented in Sec. 2.4. More rigorous formulations are reviewed in Sec. 2.5. In Sec. 2.6 the time scales of the physical effects are dis¬ cussed. Finally, basic relations from diffraction theory relevant to laser simulation are recalled in Sec. 2.7
9 10 CHAPTER 2. PHYSICS OF OPTOELECTRONIC DEVICES
2.2 Basic Semiconductor Laser Physics
A concise treatment of the general semiconductor physics can be found e.g. in [23]. The concept of the semiconductor band gap, doping, and the basic transport mechanisms are defined therein. In this introduc¬ tion, a focus is set on the physics relevant to semiconductor lasers.
In active optoelectronic devices, photons are typically generated at a p-n junction by so-called injection luminescence. Electrons and holes are supplied by a driving current and injected into the active region, which typically is an intrinsic region between the n and the p region. A necessary condition for efficient light generation is that the active region is supplied with a large number of electrons and holes.
Photons are generated by the radiative recombination of electron-
hole pairs. The angular frequency ui = 27r/ of the electromagnetic wave can be calculated from the energy difference AE of the transi¬ tion,
2ttc AE ,n^
where c is the speed of light, Ao the free-space wavelength, and h is the reduced Planck constant. Reversely, electron-hole pairs can be generated by the absorption of an incident electromagnetic wave, provided that the frequency is high enough to overcome the bandgap of the semiconductor material.
Optoelectronic semiconductor devices are typically fabricated us¬ ing compound semiconductor materials, such as AlxGa(x_i)As, or GaxIn(1_x-)AsyP(i_y). The combination of different material com¬ pounds is restricted to compositions with matching atomic lattice constant. Obeying this constraints, the bandgap of a compound semi¬
conductor can be adjusted by the material compositions x and y. In order to prevent the absorption of the photons generated in the ac¬ tive region, the bandgap is larger than the photon energy everywhere except in the active region. The design of a suitable layer structure is called 'bandgap engineering'.
There are two types of radiative recombination: spontaneous emis¬ sion and stimulated emission. In the case of spontaneous emission, photons are emitted in random directions with no phase relationship 2.3. LUMPED LASER EQUATIONS 11
among them. In the case of stimulated emission, an incident pho¬ ton initiates the recombination and generates an excess photon that matches the original in energy, phase and direction of propagation. As a result of the stimulated emission, an electromagnetic wave is amplified coherently. A device that employs Light Amplification by Stimulated Emission of Radiation is called LASER.
The amplification is accounted for by the material gain rst(r,u>), which is a local property that depends on the angular frequency u of the electromagnetic wave. It can be calculated from the local carrier densities n and p (see Sec. 2.4.3).
Introducing optical feedback allows the formation of a standing wave in the laser device. For the standing wave, an amplification factor (or modal gain) G can be calculated from the overlap of the optical mode pattern with the local material gain. The total loss L on the other hand is the sum of absorption loss, radiation through partially reflective mirrors, and scattering loss. In the absence of a driving current, gain is negative and an electromagnetic wave is absorbed in the active region. With an increasing driving current, gain becomes positive. Lasing action is achieved when modal gain meets the total losses. A further increase in the driving current results in an efficient conversion of injection current into optical output power.
An electromagnetic wave is guided by material with high refractive index. A waveguide structure typically supports a discrete number of modes. Potentially, several of the competing modes can start to läse when their individual modal gain exceeds the modal loss. Lasing action starts with a single mode and eventually comes in a multi-mode operation regime when the driving current is increased.
2.3 Empirical Laser Model: The Lumped Laser Equations
The main characteristics of semiconductor lasers can be described by two coupled rate equations. The carrier rate equation and the photon 12 CHAPTER 2. PHYSICS OF OPTOELECTRONIC DEVICES
rate equation,
dN
— - - - HI 9(N) vgS Rsp(N) Rnr(N) (2.2) dS_ Tg(N) vgS- — + TßspRsp(N), (2.3) ~dt Tph with carrier number A/ and photon number S. The driving force is the injection current /. Some portion of the current does not reach the active region, which is taken into account by the injection efficiency 0 < rji < 1. The variables Rsp(N) and Rnr(N) are the spontaneous and non-radiative recombination (unit: [1/s]). The stimulated recom¬ bination is the product of the incremental optical gain g(N), the group velocity vg and the photon number S. The stimulated and the spon¬ taneous recombination appear in both rate equations. They reduce the carrier number and increase the photon number. From the spon¬ taneous recombination, only the part that is emitted into the lasing mode contributes to the lasing power. Therefore, the term is multi¬ plied by the spontaneous emission factor ßsp, which is in the order of 10-3 to 10_1. The parameter V is the mode confinement factor, and
Tph is the photon life time. The functional relationship of the recombination and generation rate contain empirical parameters. A set of models is
Rsp(N) = B- N2 (2.4)
Rnr{N) = A- N + CN3 (2.5)
g(N) = a-(N-Ntr), (2.6)
Where A is the coefficient for the defect recombination (Shockley- Reed-Hall model), B is the spontaneous emission coefficient, and C accounts for Auger recombination [1]. The linear gain model uses the differential gain a as a parameter and contains the transparency car¬ rier number Nfr. Linear gain is only applicable in a limited range. In order to simulate effects such as the roll-over in the I-P characteristic, it has to be further extended, e.g. by a parametric dependence on the photon number 5, by a logarithmic dependence on carrier density, or 2.3. LUMPED LASER EQUATIONS 13
02 03 04
Current [A]
Figure 2.1: Laser characteristics, derived by a laser rate-equation model with two optical modes.
a combination of the two [1],
g(N,S) a-(N- Ntr) + b-S (2.7)
A/ + go , ( Ns g(N,S) In (2.8) 1 + eS Ntr + Ns
with the empirical gain coefficient go, the saturation coefficient e and the shifting carrier number Ns. While these parameters are simple fitting parameters, it is possible to include results from semiconductor physics by introducing a wavelength dependent gain model.
Steady-State Characteristics
In the steady-state, the time-derivatives in the rate equations vanish. In Figure 2.1 the solution of the photon rate equation with linear gain is shown.
For lasers, the output power can be calculated from the photon 14 CHAPTER 2. PHYSICS OF OPTOELECTRONIC DEVICES
20
Higher Output Power = Resonance m Higher Frequency + Higher Damping
£Z O Q_ (/) Z3 o -10 Frequency [GHz] Figure 2.2: Modulation response \P(uj)/Pq\. For higher output power, the resonance frequency as well as the damping is higher. density by pLas = out (i-n)^S, (2.9) q with the front mirror reflectivity r\. The I-P curve relates the laser output power to the driving cur¬ rent. It has two linear regions, separated by the threshold current Ith- Below threshold, spontaneous emission is predominant. Above threshold the light is generated by stimulated emission, and the pho¬ ton number becomes very large. The threshold gain can be calculated from (2.3), gth = (2.10) FVgTph The differential quantum efficiency r1PLas a y arout Vd (2.11) hui dl is the number of photons generated per electron that is injected into the device. 2.3. LUMPED LASER EQUATIONS 15 2 04 CD O T 3 CD 00 Figure 2.3: Transient response, calculated by the empirical laser model. The driving current is driven from below to above the thresh¬ old at time to = 0 s. For LEDs or lasers in the sub-threshold regime, there is very small stimulated recombination, g(N)S ~ 0. The rate equations are signifi¬ cantly simplified and the output power can be calculated by yLED — Pout = VHrVc I = r]ex—/, (2.12) with the definition of the radiative efficiency r\r = Rsp/(Rsp + Rnr) and the out-coupling efficiency r\c. For LEDs, the product r\ir\rr\c is the external quantum efficiency r)ex. Transient Analysis and Modulation Response From the rate equations, the small-signal frequency response can be obtained analytically. The transfer function and the resonance fre- 16 CHAPTER 2. PHYSICS OF OPTOELECTRONIC DEVICES quency for linear gain are [1] P(u) VTphUJ. (2.13) JM {{^-^rf+^{l/T + Tph^)2) u2 = ^vaS/rph. (2.14) Example results are shown in Figure 2.2. On the other hand, the large-signal response can be calculated by numerically solving the rate equations (2.2) and (2.3). An example for the turn-on characteristics is shown in Figure 2.3. Note that the turn-on characteristics are instructive for determining the empirical parameters from measure¬ ments. Electrical Diode Characteristics In order to include the electrical characteristics of the semiconductor laser diode, additional models are required. A simple lumped diode model that applies for p-n diodes is I = I0(ev'v*-l), (2.15) with the leakage current Jo and the thermal voltage V*. However, this model is not sufficient to describe laser diodes accurately. At the laser threshold, for example, the I-V characteristic exhibits a kink because above threshold, the total series resistance of the diode is reduced. Due to the additional radiative recombination, more current can flow through the device. A small-signal conductance can be defined by 9ä = ~ = %- (2.16) rd dV The dynamical characteristics can be included by an equivalent cir¬ cuit model, e.g. by the inclusion of a diffusion capacitance Cdiff or a depletion-layer capacitance Cdepi- 2.3. LUMPED LASER EQUATIONS 17 Spectral Linewidth of a Semiconductor Laser In the framework of the lumped laser equations (2.2) and (2.3), the laser linewidth can be derived for edge-emitting lasers [24]. The elec¬ tric field is expanded as E{z,t) = Eoy/SWe^e^*-^, (2.17) with an amplitude Eq \/S and a phase = Au;FWhm ~17F~- (2.18) Evaluation of the Empirical Laser Model The empirical laser equations are well suited for fitting measurement data. The quality of an empirical model is high if the deviation from the measurement is small and at the same time, the number of fitting parameters is small. New observations in the measurement often re¬ quire an extension of the rate equation models (e.g. the roll-over in the I-P curve can be explained by introducing a new dependence of the gain on the photon number, as shown by the replacement of (2.6) by (2.7) or (2.8)). Electrical, thermal and optical models are usually treated separately. Some relations between lumped-element laser parameters and the device geometry have been derived, as for example the dependence of the laser threshold on device length and mirror reflectivity [1]. How¬ ever, the lumped laser model is limited because the spatial dependence of the variables is neglected. Many engineering tasks are in the op¬ timization of device details, as for example the ridge shape for edge emitting lasers [26]. For these tasks an approach based on the eigen- mode laser equations is appropriate. 18 CHAPTER 2. PHYSICS OF OPTOELECTRONIC DEVICES 2.4 Eigenmode Laser Equations In contrast to lumped element calculations, multi-dimensional device simulation accounts for the spatial dimensions of the device. Electron transport as well as heat flow are modeled in detail. In the following, the eigenmode laser equations are presented. The fundamental assumption is that the optical field can be expanded into eigenmodes [1, 27, 28, 29, 30]. The laser eigenmodes are mathemat¬ ically defined in Chapter 3 and a numerical solution is presented in Chapter 4. As a consequence of the expansion into eigenmodes, the optical field is treated in frequency-domain. The electro-thermal behavior of the device, by contrast, is a slow process (a comparison of the time- scales is given in Sec. 2.6) and is modeled in the time-domain. An analysis of the approximations made by the eigenmode laser equations has been presented in [31]. 2.4.1 Electro-Thermal Model The differential equations governing the electro-thermal device physics are the Poisson equation, the continuity equations for electrons and holes and the heat diffusion equation, V • eV0 = -q (p - n + 7V+ - N^) (2.19) V-jn = q{R + dtn) (2.20) -V-jp = ) (2.21) V • KthVT = CtatdtT - V • (jn (PnT + $n) + jp (PpT + $„)), (2.22) with electron and hole current jn = -q {finnV(f> - DnVn + finnPnVT) (2.23) Jp = ~q (/*PPV0 + DpVp + fippPpVT). (2.24) The variables are defined in Appendix D, and a comprehensive set of equations is presented in Appendix C. The basic assumption is that the charge carriers are in so-called quasi-equilibrium [3]. 2.4. EIGENMODE LASER EQUATIONS 19 The recombination rate R is the sum of the non-radiative recom¬ bination Rnr, the spontaneous recombination Rsp and the stimulated recombination Rst, R= Rnr + Rsp + E„/?**. (2.25) A positive Rst is related to photon generation while for Rst < 0 optical light is absorbed and electron-hole pairs are generated. 2.4.2 Optical Model In semiconductor lasers, the optical field and the electronic system are closely coupled. In the eigenmode laser equations the total optical field is separated into the optical field pattern \I/ and the photon number S. The device equations are extended by the photon rate equation dtSv = {Gv - Lv) Sv + ßvRsp. (2.26) The difference between modal gain Gv and modal loss Lv is the am¬ plifying factor of the photon number. The spontaneous emission cou¬ pling coefficient ßv is the portion of the spontaneous emission R^p that couples into the lasing mode, as discussed below. While (2.19)-(2.22) are partial differential equations with variables depending on the spatial coordinates, the photon rate equation is an ordinary differential equation containing only the time derivative. Derivation of the Photon Rate Equation Starting point of the derivation of the photon rate equation are the time-domain Maxwell equations (A.1)-(A.4). In order to eliminate the fast time-scale the expansion E(r,t)=E0Y/^Arit)y/S^)e-i^te-^^+c.c. (2.27) V is used. The constant E0 is chosen such that the energy stored in the optical field inside of the cavity is Wopt = TiooS. In this nota¬ tion, parametric dependence is separated by a semi-colon from the functional dependence. As an example, for the optical mode pattern 20 CHAPTER 2. PHYSICS OF OPTOELECTRONIC DEVICES ^v(r;i), there is no net contribution to the time dependence because the normalization /// (er(r; i)tf„(r; i)) *M(r; t)d3r = SVft, (2.28) holds for all times t [32]. Note that for open cavities, orthogonality of eigenvectors is not a trivial issue [33], and the integration bounds have to be restricted to the cavity volume. The introduction of absorbing boundaries in the eigenvalue problem and the theoretical analysis by using quasi-normal modes are discussed in Sec. 3.4.2. The eigenmode laser equations can be derived as follows: First, substitute (2.27) into the Maxwell equations. After this, multiply by the complex conjugate of the field and integrate over the cavity volume. Using (2.27), an arbitrary optical field can be expanded. The modal field ^y and the frequency ujv can be found from the solution of an eigenvalue problem. In principle, one is free in the choice of the basis functions ^v and its related eigenvalue problem, as long as the fields ^v are complete enough to expand any electric field in the interior of the laser [31]. The right choice of the eigenvalue problem will cancel out many terms, in particular when material dispersion is considered, i.e. when the permittivity explicitly depends on the frequency. In the following, the cavity eigenvalue equation is used, with the eigenpair {ujv, tyv) and the complex eigenfrequencies G R. ujv =Jv + iw'l , w^w" (2.30) Comparing the proposed eigenvalue problem (2.29) to the standard cavity problem discussed in Sec. 3.2.2, (V2 + ^r«) E(r) = 0. (2.31) 2.4. EIGENMODE LASER EQUATIONS 21 The difference to the standard cavity eigenvalue problem is due to the dispersive material. The cavity equation can be re-written in the form (v2 + ^ (er(r) + efsp(r,^))) *„(r) = 0 (2.32) ef>v(r^) = i^d-^±. (2.33) Defining £*ot = er + eflsp allows to treat (2.29) with an eigenvalue solver designed for the solution of the standard optical eigenvalue problem. In consequence, a series of standard eigenvalue problems (2.31) must be solved iteratively, and £*ot has to be evaluated re¬ peatedly. For typical semiconductor materials, a target value for the uj pa can be obtained from the eigenvalue problem, ujv , non-dispersive cavity due to \ejlsp\ dtSv = %JISV + Rsp. (2.34) Comparing this result with the photon rate equation from the lumped element formulation (2.3), the ratio between imaginary and real part of the eigenfrequency can be expressed by the modal gain Gv and the total loss Lv, 2uj'1 = (Gv - Lv). (2.35) The modal gain can be calculated from the local material gain rst as discussed in Sec. 2.4.3. Spontaneous Emission The remaining term in the photon rate equation (2.26) is the sponta¬ neous emission. It can be factorized into a spontaneous emission cou¬ pling term ßv and an overlap integral between the local spontaneous emission rsp and the optical mode pattern [34]. EELs and VCSELs 22 CHAPTER 2. PHYSICS OF OPTOELECTRONIC DEVICES require different overlap integrals, KP = ffrsp{r,E)\E=hœi\^\2d2r (2.36) KP = fffrsp (r, E)\E=hui |*„|2d3r, (2.37) with the transverse waveguide mode <&v and the cavity mode ^>v, re¬ spectively. Note that the spontaneous emission is frequency dependent and has to be evaluated at the lasing frequency u)\. In the following, the derivation of the spontaneous emission cou¬ pling factor ßv is presented. Starting again from the Maxwell equa¬ tions (A.1)-(A.4), the wave equation can be derived by eliminating the magnetic field H. Since the spontaneous emission is an incohérent source in the optical field, it appears as an inhomogeneity F(r,t) in the wave equation, V2E(r,t) - f E(',*) +47T / = F(r,t). -2^2 X(r,t-r)E(r,t)dTJ (2.38) It is useful to define the Green's tensor Q to reconstruct the electric field from an arbitrary source F(r,t), - —— / V2£(r,i,r',i') f 0(r,t,r',t') + 47r X(r,t-r) Q(r,t,r',t') dr J = S(r-r',t-t') (2.39) t E(r,t)= /// / g(r,t,r',t')F(r,t)dt' d6r'. (2.40) Note that in contrast to the scalar Green's formalism, the excitation S has an orientation. The dyadic analysis can be applied for a compact notation [35, 36]. Due to the stochastic nature of F, neither averaging in time nor a Fourier transform reveals the spectral contents relevant to the light generation [31]. This information is contained in the power spectrum P(uj,r). It is well known that P can be obtained from the field- field correlation function if) by a Fourier transform (Wiener-Kinchine 2.4. EIGENMODE LASER EQUATIONS 23 theorem) [37], 1 pt+T/2 Ht, r,t) = - E(r, t' + r/2) • E* (r, t' - r/2) df (2.41) 1 Jt-T/2 ±- P{u, r;t) = J(iP(t, r, t)) e~^dr, (2.42) where the ensemble average denoted by (.) is taken over a time long enough to cancel out the signal contents at optical frequencies but leaving the modulation frequency intact. Consequently, the time- dependence in the power spectrum P(uj, r; t) is 'on the electronic time scale'. Inserting (2.40) in (2.41) and the result in (2.42) reveals the de¬ pendencies between an arbitrary source F and the power spectrum measured at a position r in the device. With the expansion of the electric field into eigenmodes, the spontaneous emission coupling into a particular mode v can be calculated. A numerical calculation of the spontaneous emission coupling factor ßv for an EEL is presented in Sec. 5.2.2. 2.4.3 Local Electron-Photon Interaction In addition to the global variables Gv, Lv and Rsp, the local interaction between optical, electrical and thermal effects have to be included to close the opto-electro-thermal feedback loop. Gain Gv and total spontaneous emission Rsvp in the photon rate equation are calculated by integral relations containing the local mate¬ rial gain rst(r, E) and the local spontaneous emission rsp(r, E). Both can be calculated from the local carrier densities. Furthermore, the electron-photon interaction influences the com¬ plex refractive index n in its real and imaginary part, = = - - n n' + in" n'bg + An'th + An'st + An'fca £- (rst a*ca) . (2.43) The real part n' is composed of a background refractive index n'b and corrections An[ due to temperature dependence, stimulated emission 24 CHAPTER 2. PHYSICS OF OPTOELECTRONIC DEVICES and free carrier absorption. The imaginary part directly contains lo¬ cal material gain rst (inter-band transitions) and free carrier absorp¬ tion (intra-band transitions) œca, while the correction terms An'st and An'r are the consequence of the Kramers-Kronig relation, as discussed below and in Appendix B. The temperature dependence can be expressed by Arith = ath AT. (2.44) An increase in lattice temperature focusses the optical field. The term ath > 0 can be determined empirically. In semiconductor lasers, the dependence of n on the local electric field is not relevant. In operating condition, the laser is a forward- biassed diode, and the local electric field is not strong enough to cause a significant change of the refractive index. However, field dependence is important for modulators [38]. Radiative Recombination Stimulated and spontaneous radiative recombination are needed to calculate the local material gain rst and the local spontaneous emis¬ sion rsp, respectively. Radiative recombination can be understood microscopically by the resonance between the oscillating electromagnetic field (photons) and the charge carriers (electrons). In a semiconductor laser, the quantum mechanical transitions typically occur between an initial state and a continuum of final states. As a subsequent process, the distribution functions relax due to carrier—carrier and carrier—phonon scattering (dephasing of carriers in k-space). In consequence, radiative recom¬ bination as well as photon absorption are irreversible processes. This is in contrast to a two-level system where energy tends to oscillate between the two states. For the application in optoelectronic device modeling, it is not necessary to quantify the time evolution of the photon-carrier inter¬ action. Fermi's Golden Rule [39] can be applied to get an expression for the transition rate of the radiative recombination. Local material gain rst and local spontaneous emission rsp can be calculated from 2.4. EIGENMODE LASER EQUATIONS 25 electron and hole distribution functions fc and fv, respectively [3], rst(r,E) = E/ ColM^lVW') (/fW#) + fJ(r,E>) - l) LC^E'jdE' (2.45) rSp(r,E) = E/ ColM^-lV^E') (/f (r,E')//(r,E')) L(E,E')dE'. i,3 (2.46) Electrons and holes are assumed to be in quasi-equilibrium and the quasi Fermi potentials <3>n and $p are introduced, fC = (1 + e^(Ec+q*n+^E)y1 {2A7) fv = (i + e^v+q%+^eE)y\ {2M) Electron and hole densities are related to the quasi Fermi potentials by > = »M*ar")' (2'50) where .F1/2 is the Fermi integral of order 1/2 [3]. The definition of all parameters can be found in Appendix D. The use of Fermi poten¬ tials in semiconductor device simulation is discussed in [40] and the implementation of the above equations is presented in [41, 42]. While the material gain directly influences the imaginary part of the refractive index, the influence on the real part has to be considered as well. Since rst(r, E) is known for the entire frequency (or, energy) range, the Kramers-Kronig integrals can be applied, An>st(r,E) = ^V I '~^dE. (2.51) e2 Jo E2 - E2 A numerical integration of this integral has been presented in [43]. A considerable local refractive index change (\An't | « 0.1...0.2) has been 26 CHAPTER 2. PHYSICS OF OPTOELECTRONIC DEVICES found for InP material with carrier densities ranging from 1016 cm-3 to 1019 cm-3. An additional effect of of the same order of magnitude is expected due to bandgap shrinkage. For semiconductor lasers with a quantum well active region, there is only a very small region with a bandgap below the photon energy. The influence on the optical mode pattern is expected to be small, but further investigations in this area are required. Free Carrier Absorption In addition to the emission or absorption of a photon by the recombi¬ nation or generation of electron-hole pairs, a photon can be absorbed by the excitation of an electron from the conduction band edge to a higher energy level in the conduction band. Due to the fast electron- electron scattering, a quasi-equilibrium is obtained with an increased electron temperature. Finally, electron-phonon scattering heats up the crystal lattice locally. The processes are modeled in the first-order Drude model [44, 45]. Applying the Kramers-Kronig integrals (see Appendix B) [46, 43], the contribution to the real and imaginary part of the refractive index are A^ = -8Ä%^t + ^J (2-52) a*« = ^An'Jca = -#- (-^ + -E^\ /ca , (2.53) c 47r2c3e0n \fiem2 Hhtn2hJ using standard definitions (see Appendix D). Note that the permittivity er = n2 appears in the wave equa¬ tion (2.31). Squaring a complex number, real and imaginary part are mixed. In summary, carrier injection reduces the real part of er and causes a de-focusing of the electromagnetic wave. 2.4.4 Device-Specific Models The formulas for the calculation of the lasing frequency, the modal gain and the modal loss are specific for the different device types. Both FP and DFB/DBR lasers have an extreme aspect ratio with a device length £ much longer than the cross-section diameter. The 2.4. EIGENMODE LASER EQUATIONS 27 optical modes can therefore be separated into a transverse 2D eigen¬ value problem and a longitudinal ID solution. The transverse equation is the same for FP and DFB/DBR lasers, (v2 + ^er(r)-72) Fabry-Perot Laser For Fabry-Perot lasers, the long cavity supports a large number of optical modes. In the spectrum, the modes are dense and compete with each other. The lasing mode is the one with maximum gain, ui = max J rst(r,hu)\<5>v(r)\2d2r. (2.55) The modal gain is the weighted local material gain, evaluated at the lasing frequency, Gv= IJ' rst{r,E)\E=huji\ The cavity loss is the sum of the mirror loss due to the finite mir¬ ror reflectivities and the waveguide loss obtained by the transverse eigenvalue problem Z.„ = 7ln +Im{lv), (2.57) where £ is the laser length and r\, r2 are the amplitude reflectivities of the front and back mirror. Note that the power reflectivities are the square of the amplitude reflectivities. Edge-Emitting DFB/DBR Laser For DFB/DBR lasers there are only a few optical modes in the relevant frequency range. The angular frequency uj is determined by the cavity. 28 CHAPTER 2. PHYSICS OF OPTOELECTRONIC DEVICES A ID longitudinal cavity problem has to be solved, (v2 + ^er(*)W(*) = 0. (2.58) The gain has to be evaluated at the resonance frequency, and conse¬ quently, the devices are not necessarily driven at the gain peak. VCSEL In contrast to EELs, there is no possibility for VCSELs to employ a separation into a longitudinal and a transverse direction. As a conse¬ quence, the cavity equation has to be solved, (V2 + ^4°*(r)) *v(r) = 0. (2.59) The resulting eigenvalue uv = uo'y + i uj" holds all the information about gain, loss and the lasing frequency. 2.4.5 Extension of the Eigenmode Laser Equations 3D Simulation Even if the structure is homogeneous in longitudinal direction, there is a considerable longitudinal dependence of the optical fields. Fabry- Perot lasers typically have an anti reflective (AR) coating at the front mirror and a highly reflective (HR) coating at the rear mirror. The envelope of the optical intensity grows towards the front mirror. In the longitudinal variation of DFB lasers, sharp peaks appear when phase shifts are introduced [9]. In consequence, the structure has to be simulated in full 3D. An example of the 3D optical field in a DFB laser is presented in Figure 2.4. Multi-section lasers show a different 2D behavior for each section, and the optical field changes at least in the transition region between the different sections. Correction Term for 2D EEL Simulations For edge-emitting devices, a simple correction term to the spontaneous emission takes into account that the longitudinal field distribution is 2.4. EIGENMODE LASER EQUATIONS 29 Back Mirror Front Mirror B 0 0 200 400 Distance Longitudinal (z-Axis) [|xm] ^ Top Contact--""" Regrowth Region Buried Confinement- Active Region --""' Back Mirror Substrate --""" ' DFB Grating Bottom Contact Light Output Front Mirror Figure 2.4: Simulation results of the optical field in a DFB semi¬ conductor laser. The structure consists of a buried confinement for the optical wave and the current. The longitudinal grating is dis¬ played schematically. The device facets provide additional optical feedback due to the finite reflectivity of the semiconductor-air inter¬ face (R « 27 %). The envelope of the optical field in longitudinal direction is shown in the inset. not constant. It has been explained above that the laser intensity drops towards the center of the laser. As a consequence, the stimu¬ lated emission is lowered near the center of the laser while the spon¬ taneous emission is evenly distributed along the length of the laser. Using the stationary solution of the photon rate equation (2.64), 5 = Rsp ' (2.60) L-G the photon number S can be expressed as amplified spontaneous emis¬ sion. Therefore, it has been proposed [47] to account for this longitu¬ dinal effect by a spontaneous emission enhancement factor called the 'Petermann Factor', f\E2(z)\dz Kz = (2.61) fE2{z)dz 30 CHAPTER 2. PHYSICS OF OPTOELECTRONIC DEVICES where E(z) is the longitudinal electric field distribution and the in¬ tegration is taken along the entire cavity. The modified photon rate equation is dtSv = vg (Gv - Lv) Sv + vgßvKzR8J>. (2.62) While Kz has originally been proposed for gain-guiding lasers [48], it has been shown later that it is also applicable for index-guided lasers with low facet reflectivities [34]. To put (2.61) into words, Kz increases for cavities with high losses. For Fabry-Perot cavities, + Vîh)(i-VWïh\ = ((^h K ' " V v/?57kln(l/(ßiß2)) J Note that with this choice, the calculation of the transverse optical field in a 2D simulation directly gives the field strength and power flux at the mirror facets. For a 3D simulation the longitudinal inho- mogeneity of the optical field and the gain is considered rigorously, and it is not necessary to introduce Kz. 2.4.6 Beyond the Adiabatic Approximation Recently, an extension of the adiabatic approximation has been pro¬ posed [31]. It is based on the eigenmode expansion of the optical field as described in the preceding section. In contrast to the standard derivation, the time derivative of the dielectric function is retained. This results in additional terms in the derived equations. It is shown that their effect can be accounted for by introducing an additional coupling between the adiabatic modes, dtSv = (Gv - Lv) Sy + R8J + Ev. (2.64) The new term "Ev accounts for photons that were emitted spontaneously at earlier times into other modes, have been amplified there, and are now leaking into the mode of interest, Sm(*) = E / RtP(r)T^(r,t)dr, (2.65) J — .. oo 2.5. MORE FUNDAMENTAL LASER MODELS 31 where T can be calculated from the cavity Green's functions. T is oscillating rapidly and it can be seen from (2.65) that SM(£) vanishes in the adiabatic approximation. The integral over the entire history of the laser is a problem for the calculation of this correction. As a remedy, a recursive algorithm is presented in [31]. 2.5 More Fundamental Laser Models In principle, the carrier dynamics for inhomogeneous semiconductor devices takes place in real space and in ic-space. Furthermore, the physical processes happen on the sub-femto-second time scale [49, 50]. From the modeling perspective, a rigorous numerical treatment be¬ comes computationally intensive. Some simplifications are necessary in order to calculate steady-state device characteristics. A hierarchy of approximations is presented below. In consecutive order, the range of the models can be summarized as follows. • Assume parabolic bands and expand the energy axis to a finite number of allowed states for the electrons. A problem in this formulation is to describe the many-body scattering efficiently and accurately [51, 52, 53]. • Introduce a two-level atomic system but solve the system on a fast time scale. As a result, the semiconductor Bloch equations are obtained (Sec. 2.5.1). • Approximate optical waves by a slowly varying amplitude. The system can then be solved on an electronic time scale (Sec. 2.5.2). • Expand the optical wave into eigenmodes and integrate over the cavity. The coupling between optics and electronics is then achieved by a photon rate equation (Sec. 2.4). • Reduce the electronic system to a finite number of idealized lumped elements. As a result, one gets a system equivalent to the empirical rate equations (Sec. 2.3). Furthermore, there have been investigations that specifically focus on one of the effects, such as a microscopic description of the semi- 32 CHAPTER 2. PHYSICS OF OPTOELECTRONIC DEVICES conductor without attempting to resolve the device structure in real space [54, 55]. 2.5.1 Semiconductor Bloch Equations A good compromise for the approximation of the processes in lc-space is the two-level atomic system. A solution on the fast time scale results in -^ - - ^^ dtn(z,t) = ±E(r).dtP(r) (2.66) q Tiuj r dtH(r) = --V x E(r) (2.67) dtE(r) = -\- (V x H(r) + P(r)) (2.68) dfP{r, t) + adtP(r, t) + uj2P{r, t) = An(r, t). (2.69) In addition, the semiconductor equations (C.2)-(C4) can be used to calculate the current flow and the carrier density n(z,t). An early attempt to implement a comprehensive model (including electron and hole temperature) and its application to VCSEL simulations have been presented by [56]. The semiconductor Bloch equations have been applied for the sim¬ ulation of broad area lasers [57, 58], for the analysis of the polarization processes in VCSELs [59], and for defect-mode photonic crystal lasers [60]. In all three cases, the spatial domain has been restricted to the 2D active region, and an effective-index type approximation has been made for the remaining spatial dimension. The so-called 'effective Bloch equations' have been presented [52] to overcome the errors introduced by the two-level model for the atomic transition. It is proposed to use a superposition of several parameterized Lorentzians instead of a single two-level system. In a pre-processing step, a lookup table is created from the microscopic theory for the parameters in the new model. 2.5.2 Slowly Varying Amplitude Approximation The disadvantage of the semiconductor Bloch equations is that the stiff system of equations has to be solved on the fast time scale im- 2.6. CHARACTERISTIC TIME SCALES 33 posed by the Maxwell equations. This can be avoided if the structure under consideration is essentially a ID problem. In consequence, it is possible to expand the optical field into = + E(r,t) E0 -^ - - ^^ dtn(z,t) = rst(z;t) (Ef(z,t) + Eb(z,t)) (2.71) q t dtEf{z,t) = c dzEf{z,t) + a(z) Ef{z,t) + ß^E- (2.72) n(z ti dtEb(z,t) = c • dzEb(z,t) + a(z) Eb{z, t) + ß-^-L. (2.73) The analysis is limited to structures where phase fronts are planes perpendicular to the propagation direction. However, for the analysis of longitudinal mode competition in lasers or the behavior of semi¬ conductor optical amplifiers, these models can be very useful. 2.6 Characteristic Time Scales A separation of fast and slow processes, as presented in Sec. 2.4, re¬ quires a detailed analysis of the time scale on which the physical pro¬ cesses take place [61]. Different carrier reservoirs are identified in Fig¬ ure 2.5. The energy exchange rate between the reservoirs is discussed in the following. 1. The characteristic time scale for the driving current of a laser is of particular interest in telecommunication applications. It can be stated as a modulation requirement or the maximum allowed turn- on / turn-off time of the device. Today's maximum performance is in the region of 40 GHz modulation of the applied signal. For future devices, this may be extended to 100 GHz or a typical time constant ofrin = 10-11s. 2. The device electronics cannot be infinitely fast since it essentially follows a drift-diffusion relationship. As a consequence, reducing the 34 CHAPTER 2. PHYSICS OF OPTOELECTRONIC DEVICES length scale makes devices faster. A lower limit for the device size is the optical wavelength. From this length scale, the mobility a and the diffusion constant D, a typical time constant is rdd = 10"12 s [62]. 3. In the non-radiative recombination, a phonon is involved in the elec¬ tronic transitions. Energy is dissipated as heat in the semiconductor crystal lattice. In a real semiconductor material, a small number of excess states are present inside the energy gap due to defects and impurities, and surface states. All these effects can be treated with the Shockley-Read-Hall theory. Furthermore, even in a perfect bulk semiconductor, scattering mechanisms involving phonons have to be expected in the form of Auger recombination. Due to the collision between two electrons, one electron recombines with a hole and the other is excited to a higher energy state in the conduction band. The high-energy electron eventually thermalizes and thus gives its excess energy back to the crystal lattice. For typical laser applications, the non-radiating recombination time is of the order of rnr = 10"10 s [1]. 4. Once heat is generated, it spreads inside the device following a dif¬ fusion law. Usually, the devices are mounted on a good heat sink and the heat transfer can be modeled by simple equivalent circuits involving a thermal resistor and a thermal capacitance. Similar to the drift-diffusion case, the characteristic length scale has to be con¬ sidered to calculate the time constant Theat = 10"10 s. Note that the time constant for the vicinity of the active region is smaller than the overall thermal response of the device. 5.-7. The microscopic description of the interaction between charge carri¬ ers includes scattering processes in momentum space. Both quantum well scattering as well as spectral hole burning fall into this category. Detailed description of these processes is given in [63] and [64], re¬ spectively. From the analysis of spectral hole burning for InGaAs QW laser at 20 mW output power [65, 66], the spectral hole has an effective width AE = 0.05 eV. The scattering time is between rSCat = 10"14 s and rscat = 10"13 s. 8. The two kinds of radiative recombination are the spontaneous and stimulated emission. The typical time constant of a radiative electron- hole recombination is rreCom6 = 10~10 s. 9. The decay of the optical field in a lossy cavity (i.e. the photon life time) is the characteristic measure for the optical radiation. The photon life time is proportional to the quality factor of the cavity, and for typical EEL, rph = 10"12 s while for VCSELs rph = 10"10 s. 2.6. CHARACTERISTIC TIME SCALES 35 Power Supply ©|le-11s Electric Fields -2) 1e-12s '4 © ©I QW 1c ^Q^^n^el Çh^3^®.14 1e-14s M1e-10s 3)|le-10s ifF li Phonons Photons 1e_14s *•« 5s Mie-ios 5)lle-11s "Hr * Heat Sink EM Radiation Figure 2.5: Schematic device physics of a diode laser. Boxes sym¬ bolize carrier reservoirs and arrows denote the energy exchange. The numbers correspond to the text and stand for the different processes. Typical time constants are indicated and are discussed in the text. 10. For the oscillating electromagnetic field, the typical time constant is the resonance frequency fosc or the oscillation time reimag = fö^c = 10"15 s. 11. Similar to the electromagnetic optical wave treatment are the lattice vibrations of the semiconductor crystal. A typical oscillation period 14 iS riatticevib — 10 S. The elimination of rapidly oscillating terms is applicable to all in¬ ternal energy exchange inside one of the reservoirs denoted as boxes in Figure 2.5. The approximations presented in Sec. 2.5 can be justified by a comparison of the different time scales involved. Only the rele¬ vant steps applied for the derivation of the eigenmode laser equations (Sec. 2.4) are reviewed here. For the charge carriers, quasi-stationary distribution functions can be assumed, and the electronic problem reduces to the drift-diffusion formulation (2.19)-(2.21). For the lattice vibrations, it is sufficient to introduce the lattice temperature, and the heat equation (2.22) is derived. Furthermore, eliminating the rapidly oscillating terms results in the photon rate equation and an expansion of the optical field into eigenmodes as presented in Sec. 2.4. 36 CHAPTER 2. PHYSICS OF OPTOELECTRONIC DEVICES Observation Plane Fraunhofer Approx. Valid Fresnel \ Approx. \ Valid Kirchhoff \ ^ Integral Valid \ « ». / Screen Plane (Aperture) Figure 2.6: Geometry definition for the diffraction theory. Left: Re¬ gions of validity for Kirchhoff, Fresnel and Fraunhofer approximation, in conjunction with a VCSEL structure. Right: Variable definitions for the diffraction theory. The near field is known in the 'screen plane' (= VCSEL aperture). In an integral relation, the field in the 'obser¬ vation plane' can be calculated from the near field. The reservoir of the bulk charge carriers and the quantum well charge carriers have an exchange rate that falls in the time scale elim¬ inated by the adiabatic approximation. Nevertheless, it will appear in the equations because it is the only link in the interaction chain between the electronics and the optics. Furthermore, the intra-band relaxation is slower than the carrier interaction with the quantum well. As a consequence, spectral hole burning has to be expected in the vicinity of the active region. 2.7 Diffraction Theory Electrical and thermal device characteristics are defined on the device terminals (electric and thermal contacts). The optical field, by con¬ trast, is radiating in all directions. The total radiated energy as well 2.7. DIFFRACTION THEORY 37 as the radiation far field belong to the optical device characteristics. The solution of the coupled opto-electro-thermal device equations includes the calculation of the optical near field. The radiating fields can be calculated from the near field by the diffraction theory. In the following, the hierarchy of approximations relevant to the coupling of optoelectronics is outlined. For simplicity, the theory is presented for scalar fields. An extension to vectorial diffraction theory can be found in [32]. In Figure 2.6, the general geometry used in this theory is shown and the validity range of the different approximations are sketched. The Kirchhoff Integral The general formulation of the diffraction theory assumes known fields in screen plane r' = (x', y' ,z' = 0), referring to the coordinate system in Figure 2.6. The fields are non-zero on a finite area, which is called the aperture. From Green's theorem, a formula can be derived in which the field in the volume V can be calculated from the known field on the closed surface S = dV, s(r) = (ff (s(r')n • VÇ(r, r') - Ç(r, r')n • Vs(r')) d2r'. (2.74) It can be shown that in addition to the field s, the normal derivative dns have to be considered, and the Green's function has to be chosen according to the method of images [32], i / ikr pikr\ Ç^ = (2'75) 2 (—-—)• with r = (x,y,z)T on one side of the screen and r = (x,y, —z)T on the other side of the screen. As a result, the field s(r) can be derived for all positions r from the known field s(r') in the aperture Sa by with R = r — r' and R = litl. 38 CHAPTER 2. PHYSICS OF OPTOELECTRONIC DEVICES Intuitively, this is equivalent to spherical waves with origin in the screen. In order to get the field in the volume (i.e. at some distance from the screen), the Green's functions have to be multiplied with the near field and integrated over the screen. Note that this is a rigorous derivation, and the only approximation is that the structure under consideration may not strictly correspond to the assumption of a planar screen with zero field and an aperture with a known near-field distribution. The Fresnel Approximation For an approximation of the formula, three length scales have to be considered: the aperture size L, the wavelength A and the distance from the screen z. The phase factor etkr in (2.76) can be expanded and higher order terms are neglected. The Fresnel approximation is equivalent to the paraxial approximation. It is valid near the axis, as indicated in Figure 2.6. Note that the phase difference as well as the linear decay with distance can be drawn out of the integral, k e%klz S(r) = (2.77) It can be seen from the ((x — x')2 -\- (y — y')2)/z term that the resulting wave fronts are parabolic. The Fraunhofer Approximation (Fourier-Optics) With the approximations ^ -*1 and (1+m)^ *• (278) (2.77) can be simplified to = J_£Î^>(«a+»a) ff eik(x'2+y,2yzs(r')dx'dy'. (2.79) 2m z JJS As an interpretation, it can be seen that the wave fronts are planar. This is characteristic for the regime which is generally called the 'far field'. Note that (2.77) is proportional to the Fourier transform. 2.8. DISCUSSION 39 2.8 Discussion The focus of this work is on laser device simulation. The input for a device simulator must contain structure details, and the expectation is to derive the device characteristics by a numerical model. The ap¬ proximations presented in this chapter have to be evaluated on this basis. As a result, the optimal choice of models is the eigenmode ex¬ pansion of the optical field, coupled to a drift-diffusion electronics by the photon rate equation. Modeling the electronics in time-domain and the optics in frequency-domain is a good compromise between accuracy and computational efficiency. The energy balance formu¬ lation provides the possibility to include spectral hole burning as a correction term [67]. More rigorous laser models presented in the literature are compu¬ tationally more intensive and therefore restrict the device structure to special geometries, making an optimization of the device structure questionable. However, these models are appropriate if fast time scale processes such as the polarization switching in VCSELs are to be an¬ alyzed. Due to the complexity of the resulting dynamical system the time evolution of the fields shows a chaotic behavior [50]. Measurable quantities have to be extracted by statistical analysis of the resulting fields [68]. While the eigenmode expansion of the optical field are suitable for laser simulation, a different description has to be found for LEDs. In this case, spontaneous emission is predominant and the optical field cannot be expanded into a small number of modes. The optical behavior of the device has to be calculated from a source term in the field equations. Numerical methods suitable for such a treatment are the beam propagation method (BPM), ray tracing or a full-vectorial treatment that can be realized by the finite-difference time-domain (FDTD) method [69]. Chapter 3 Optical Cavities and Waveguides 3.1 Introduction There are two interpretations of laser modes. In the analysis of edge- emitting lasers, the laser modes are associated with the transverse optical confinement, that is obtained e.g. by a ridge or a buried het- erostructure. In the following, these modes are referred to as the waveguide modes. The characteristic modal variables are the effective refractive index and the propagation loss. In addition to the waveguide modes, the longitudinal optical field has to be solved. This is trivial in case of Fabry-Perot (FP) lasers and somewhat more involved for distributed feedback (DFB) lasers. For VCSELs, a separation in transverse and longitudinal modes is not possible. For longitudinal edge-emitter analysis and VCSELs, the rel¬ evant optical modes are the cavity modes. They can be characterized by the cavity resonance frequency ujres and the decay constant a (= inverse of the photon life time rph). In this chapter, the cavity modes and the waveguide modes are presented in Sec. 3.2 and Sec. 3.3, respectively. In both cases, the discussion includes eigenvalue equation, boundary conditions, mode designation, and spectral properties. The simplest cavity example is 41 42 CHAPTER 3. OPTICAL CAVITIES AND WAVEGUIDES the ID cavity. For waveguides, the simplest case is the slab waveguide. In Sec. 3.4, the quasi-normal modes are introduced. Time-Harmonic Analysis For the solution of the optical fields, it is convenient to write the Maxwell equations in their time-independent representation, assum¬ ing an implicit harmonic time dependence e~lU)t. Consequently, all field quantities are complex. The resulting time-independent Maxwell equations (A.9)-(A.12) can be obtained from the time-domain equa¬ tions (A.1)-(A.4) by the substitution dt —> —iu). Equivalently, the time-domain equations can be transformed to the frequency-domain by the complex Fourier transformation with u = u1 + i u>". 3.2 Optical Cavities 3.2.1 Empirical Cavity Model The simplest form of an optical cavity is the Fabry-Perot resonator. It consists of two coplanar mirrors separated by a distance £. The cavity lifetime rph and its inverse, the cavity loss L, are determined by the mirror loss as well as the loss due to absorption and scattering (see (2.57) in Sec. 4.6). The desired reflectivities can be obtained by an anti-reflective (AR) or highly reflective (HR) coating at the laser facets. Furthermore, the reflectivity can be realized by a ripple in the waveguide (DBR/DFB). In general, the resulting mirror reflectivity depends on the optical frequency. While it is easy to obtain the reflection coefficient of a single mir¬ ror, the cavity behavior cannot be obtained by a simple multiplication of the measured reflectivities Ri(co) and R2(uj). The key parameters used to characterize a cavity are the cavity finesse F and the quality factor Q. In the frequency response of a cavity, a peak can be ob¬ served at each cavity resonance. The terms F and Q can be defined with respect to the positions of the peaks (resonance frequencies) and 3.2. OPTICAL CAVITIES 43 to the peak width (resonant band width), separation of resonance frequencies resonant band width resonance frequency stored energy resonant band width power loss As a reference, the ideal Fabry-Perot resonator is recalled, 2irn£ „TT ^ ,n „. 1 - y/r\r2 A(l - v/rir2) The cavity resonance frequency cores = ujq + Au; is usually written as the sum of a reference frequency ujq and the detuning Au;. The oscillating optical field decays exponentially, E(t) = E0e-^e-^wo+Aw)*, (3.4) with the photon life time rph and the decay constant a — rvh = = —. (3.5) a üüq For a general cavity, the energy decay can be used to measure the quality factor or cavity loss. The frequency response of optical cavities can be calculated from the Fourier transform of (3.4) E(uj) = —= \ Eoe'^e-^^+^-^dt. (3.6) V 2"7T Jf) The resonant line shape is M 'E^2°>-M0-A^ + (^- The full width at half maximum of the frequency distribution function is equal to u>o/Q, relating the quality factor Q to the cavity line shape. For a detailed analysis, see [32, 7]. 44 CHAPTER 3. OPTICAL CAVITIES AND WAVEGUIDES 3.2.2 Cavity Eigenvalue Equation In the framework of optoelectronic device simulation, the optical eigen¬ modes of some given structure have to be calculated. Typically, the permittivity is a function of space, s(r), and the permeability can be assumed constant, ji = fio- With the use of the constitutive relations (A. 13) and (A. 14), the vector wave equations can be derived from the Maxwell equations (A.9)-(A.12), V x E = iujfiH (3.8) src V x H = -iueE+j (3.9) Note that j = aE is already included in (A.13) because of the complex permittivity e = Sor = os'r-\-ia/üJ. The current in (3.9) is an external source term. In order to calculate the cavity eigenmodes, the source- free equation has to be considered, jsrc = 0. The equations can be further reduced and the vectorial eigenvalue equation is obtained, (3.10) The eigenvalue equation can be viewed as black-box relationship with the material distribution as an input and the eigenpair as an output, er(r) E(r), u Cavity Eqn. (3.11) Scalar Wave Equation The scalar prototype of the cavity problem is U) A + — er(r) s(r) = 0. (3.12) It is sometimes used as an approximation to the vectorial equation. Also, some concepts such as the boundary conditions can be shown in close analogy to the vectorial case. 3.2. OPTICAL CAVITIES 45 Plane-Wave Solution In case of homogeneous space, the plane wave is a solution of (3.10), E(r,t)=Eoei^t~k-r\ (3.13) Furthermore, it can be shown that plane waves form a complete ba¬ sis of (3.10) in homogeneous space. For this kind of solutions, any frequency u is allowed and therefore, (3.10) has a continuous spec¬ trum. However, the plane wave cannot be normalized, i.e. it does not contain finite energy. Therefore, it is sometimes called a non-physical solution. Together with the boundary conditions discussed below, the cavity equation (3.10) is a well-posed eigenvalue problem and the correspond¬ ing solution contain finite energy. As a result, (3.10) has a discrete spectrum, as discussed in Sec. 3.4.2. Boundary Conditions From a mathematical point of view, the most natural candidate for a boundary condition for the vectorial Maxwell equations is the per¬ fectly electric conductor (PEC). In addition, the hypothetical per¬ fectly magnetic conductor (PMC) can be imposed. On a PEC, the transverse electric field vector Et is zero, while some normal com¬ ponent En can be nonzero. On a PMC, similar conditions apply for the H field. In the scalar case, the well-known Dirichlet or Neumann boundary conditions are required. In summary, ET = 0 Perfectly Electric Conductor (3.14) Ht = 0 Perfectly Magnetic Conductor (3.15) s = 0 Dirichlet (3.16) dns = 0 Neumann. (3-17) In microelectronics, metals are usually well approximated by PEC. The conductivity at optical frequencies, however, is finite and the penetration depth of the fields is usually several wavelengths. As a consequence, imposing PEC to truncate the computational domain of a numerical calculation is done at the risk of changing the fields sig¬ nificantly. However, PEC and PMC are well suited to treat symmetry axes. 46 CHAPTER 3. OPTICAL CAVITIES AND WAVEGUIDES Physical problems in optoelectronics are inherently open domain problems, and radiation is allowed to escape to infinity. Therefore, the Sommerfeld radiation condition is introduced. In order to keep the analysis simple, the scalar cavity equation (3.12) is considered in the following. The results are equally applicable to the vectorial wave equation. For 3D problems, lim (p (dpu - iku)) = 0, (3.18) p—>oo with k = \k\. Equivalently, for a 2D system, lim (y/p {dpu - iku)) = 0. (3.19) p—»oo The Sommerfeld radiation condition is valid because at large distances p from a scatterer, the field locally behaves like a plane wave which is traveling away from the centre. For 3D free space, u{p, ation: there is 3D radiation into free space and into the substrate. In addition, the stratified medium of the lower mirror may also support 2D surface waves (see Figure 5.7 in Sec. 5.3). Note that in the far field, surface waves show cylindrical radiation properties. With the definition (3.20), the far field pattern f((j>,0) is finite only for the 3D radiation. The cylindrical surface waves decay with 1/sfp. Therefore, f((p,0) contains sharp peaks in the direction of the surface wave. For p —> oo, Dirac distribution functions appear in the radiation pattern. 3.2.3 Mode Designation for VCSELs For VCSELs, a separation of the optical fields into a ID propagation direction and a 2D waveguide structure can only be done for special- 3.2. OPTICAL CAVITIES 47 ized geometries that hardly resemble the physical devices. However, also for the general geometry, the mode nomenclature is based on the well-known guided mode solutions for circular dielectric waveguides. For air-post VCSELs, this is reasonable because in the top mirror region, the optical confinement is achieved by the mesa structure. In the bottom mirror region, however, there is no transverse guiding structure. Two levels of approximation are typically used. Laguerre-Gauss Beam Modes or LP-Modes In the paraxial approximation, the dielectric profile is assumed to be a parabolic function er(p) = £br9 — 4/?2/'(&oro)•> with a constant background permittivity, ebr9, the distance from the axis p and the pa¬ rameters ko and ro- In the special situation of the parabolic dielectric profile, all .E-field vectors are parallel and it is sufficient to solve a scalar equation. For this problem, there is an analytical solution, as described in [70, 71]. For circular symmetric structures, the solutions are the Laguerre-Gauss Beam Modes. For a rectangular cross-section, a similar procedure is followed, which results in the Hermite-Gauss Beam Modes. The modes are also called the Linearly Polarized (LP) Modes. Due to the rotational symmetry of the structure, the modes are degenerate. Two modes with orthogonal polarization have identical propagation constants. A linear combination of two mode solutions satisfies the equations. As an example, the ring modes can be derived from the eigenmodes, Malt -i- <3> — (3.21) ' fitftfil J. ** — ~ ppjp + mrm„M (3.22) For a wide range of structures, the paraxial approximation and the resulting LP modes agree well with measurements [72]. Note that the intensity pattern is measured, and with the use of a polarization filter, mode patterns similar to the LP modes are obtained. 48 CHAPTER 3. OPTICAL CAVITIES AND WAVEGUIDES Figure 3.1: Body-of-revolution expansion for VCSEL modes. Due to the circular geometry of the VCSEL, the fields can be expanded by a Fourier series. Note that this is possible for vectorial fields, while for simplicity, a scalar field is shown. EH and HE modes Solving the vectorial equations results in a different set of eigenmodes. Typically, one particular mode in the paraxial approximation is seen to be a collection of degenerate modes in the rigorous analysis. The degeneracy is removed if higher order corrections are taken into ac¬ count. Therefore, the nomenclature derived from the Laguerre-Gauss beam modes is not sufficient for the labeling of the rigorous solution of a circular structure such as a VCSEL. The vectorial solution of the dielectric waveguide are the EH and HE modes [73]. The correspon¬ dence between the EH/HE modes and the LP modes can be found in [72]. Body-of-Revolution The VCSELs under investigation are rotationally symmetric in the z- axis. In consequence, the fields can be expanded into a Fourier series 3.2. OPTICAL CAVITIES 49 (Body-of-Revolution (BOR) expansion [74]), E = J2 (El{p,z) + EZ(p,z)e+) eiv+ (3.23) V = J2 (Ec(p, z) cos(v V where (fr is the azimuthal angle and v is the order of the Fourier term. The BOR expansion reduces the numerical problem from a 3D prob¬ lem to v 2D problems, and is therefore often labeled as 2.5D. The BOR expansion can be applied to several formulations of the optical problem. The BOR expansion allows an exact treatment of the Maxwell equations. The maximum order of the expansion is determined by the source distribution of the problem. Applications include the simula¬ tion of dipole radiation in microcavities [75, 76, 77] and the calculation of VCSEL modes [78]. For the calculation of cavity eigenmodes, the BOR expansion is perfectly suited since no source term is present that has to be ex¬ panded into a (potentially infinite) sum and the cavity eigenmodes themselves conform to the Fourier expansion. The BOR expansion is equally applicable for the different discretization methods such as the FDTD scheme (Sec.4.3.1, [79, 76]) or the edge-element method (Sec.4.5, [80, 81]). For the coupled opto-electro-thermal simulation, it is intuitive to assume ring modes for the optical pattern and no azimuthal varia¬ tion of the carrier density and temperature. However, some small deviation from the circular symmetry is always present. It removes the degeneracy of the eigenmodes. As a result, one of the modes dominates and takes all the gain, and there are no ring modes in a single-mode VCSEL. Due to the coupling between the optical and the electro-thermal system, the carrier density and the temperature are not strictly rotationally symmetric. Therefore, the 2D model is an approximation to the real 3D situation. 3.2.4 ID Cavity The rigorous solution of the 2D or 3D cavity eigenvalue equation is presented in [81] and results are shown in Sec. 5.3. The ID case needs 50 CHAPTER 3. OPTICAL CAVITIES AND WAVEGUIDES 4 — 3 x 2 » > "G £ 1 o -i—« Q. OH O M— 0 O M ro Q. "S -5-4-3-2-1 0 1 2 3 4 Distance z [|im] Figure 3.2: Optical eigenmode of a ID cavity. The optical resonator is formed by two distributed Bragg reflectors. The mirrors are formed by A/4 layers of AlGaAs with varying mole fractions. The resulting variation of the refractive index is shown in the top curve. Below, the real part of the resulting optical eigenmode is shown. further discussion. It is an important special case of open resonators and can be applied for the approximative solution of the VCSEL prob¬ lem. Furthermore, its efficient solution is an important ingredient in the 3D simulation of edge-emitting lasers. As shown in Figures 3.2 and 3.3, the main characteristic of the optical fields can be obtained from a ID analysis. Inside the laser cavity, the fundamental optical mode is a standing wave with the typical node and anti-node behavior. The cavity is called an open resonator because some portion of the optical field is radiating from the cavity and escaping to infinity as a traveling wave. The formulation with counter propagating waves allows to avoid a solution of an algebraic eigenvalue problem. Instead, the eigenpair can be found by a root search in the complex plane (see Sec. 4.3.2). The ID wave equation in the time-domain reads d2zE{z,t) - £-^-d2E(z,t) = F(z,t). (3.25) 3.2. OPTICAL CAVITIES 51 Several expansions are possible. For general multidimensional struc¬ tures, the time-harmonic analysis is applied to remove the time-depen¬ dence in the wave equation (resulting in equation (3.10)). In con¬ trast to this, the longitudinal field of a ID structure can be split into rapidly varying forward and backward propagating waves E+(z,t) and E~(z,t), respectively. The fast variation in time and space can be separated from the field amplitude, resulting in the slowly varying fields E+(z,t) and E~(z,t). In summary, E(z, t) = E(z)e-iwt + c.c. (3.26) = Ë+(z,t) + Ë-{z,t) (3.27) = E+(z,t)e-iw»t+iniz + E-{z,t)e-i^t-iniz. (3.28) Since the laser cavity is an open resonator, the Sommerfeld radiation condition is required, dzE(z)-i-elreftE(z) = 0 z<0 (3.29) c dzE(z) + i-e?9htE(z) = 0 z > £, (3.30) c where the problem is defined on the interval [0,1]. No index step is allowed outside of this interval and elre^, err%9ht are the constant permittivities on either side of the laser. In words, the Sommerfeld radiation condition only allows light traveling away from the structure and prevents incident waves. In a calculation considering right and left traveling waves, the Sommerfeld radiation condition is met just by the introduction of the finite mirror reflectivities r\ and r dlE(z) + ^E{z) = F(z). (3.31) An example of a ID VCSEL cavity is shown in Figure 3.2. On a GaAs substrate, AlGaAs and InAlGaAs are grown subsequently to form a 40-layer bottom Bragg mirror, a A-cavity, and a 20-layer top Bragg mirror. The resulting refractive index n(z) = \/e(z) is shown in the upper part of the graph. Below, the resulting eigenmode Re (E(z)) 52 CHAPTER 3. OPTICAL CAVITIES AND WAVEGUIDES Figure 3.3: Reflection Coefficient of two layer structures. The struc¬ tures are sketched on the right of the diagram. While Structure A consists of a single Bragg stack, Structure B is an optical cavity, sandwiched by two Bragg mirrors. In the diagram, the dotted and the continuous line correspond to the reflectance of Structure A and B, respectively. is displayed. In the air region (z > 2.6 //.m), the laser output can be seen. The solution has been obtained by the transfer matrix method, as presented in Sec. 4.3.2. 3.2.5 Spectral Properties In the following, the cavity spectrum is discussed in detail and a ref¬ erence is made to cavity measurements. Cavity Spectrum In the theoretical analysis of optical cavities, it is common to charac¬ terize optical cavities and waveguides by the indication of the poles in the complex u plane. This is possible because for every source point r in the cavity, and for every observation point r', the Green's function Q(r,r',uj) have the same poles, each pole belonging to an eigenmode of the cavity. A point source probing the cavity with a delta function in time couples to all eigenmodes, except to the modes that have a node at the source location. The response taken at any position in the cavity 3.2. OPTICAL CAVITIES 53 (i.e. the Green's function Q) reveals all cavity resonances. The dis¬ crete set of poles is called the spectrum of the cavity. For practical applications, it is sufficient to indicate the poles near the resonance frequency. Illumination Experiment For the measurement of the spectrum, the source as well as the probe are placed outside of the cavity. The cavity is illuminated from outside and the reflectivity reveals the cavity resonance. In Figure 3.3, reflec¬ tion coefficients are shown for a stack of Bragg mirrors (Structure A) and for an optical cavity of thickness A, sandwiched by two identical Bragg mirrors (Structure B). It can be seen that a VCSEL cavity be¬ comes transparent at the Bragg resonance frequency, while an isolated Bragg stack does not show the sharp dip at the resonance frequency. The quality factor Q and the decay constant a can be calculated from the spectral line width of the resonance by using (3.5). A VCSEL structure contains an active region in the center of the cavity. In the active region, photons can be absorbed or generated, depending on the injection (or pump) current. Illuminating an un- pumped VCSEL device reveals the sum of the radiation loss and the absorption loss, atot = r~^ = aabs + arad. In order to measure the radiation loss separately, special VCSELs without active region have to be fabricated. Alternatively, the VCSEL can be pumped electron¬ ically to achieve a transparent active region [82]. In the reflection coefficient R(uj), it can be observed that the dip at the resonance frequency gets broader for cavities with larger loss. This must not be misinterpreted as a transition to a continuous spec¬ trum. The cavity spectrum always remains discrete, including open and absorptive cavities. Reducing the mirror reflectivities increases the imaginary parts of the poles. The broadening in R(uj) is equivalent to the shift of the poles from the real axis. 54 CHAPTER 3. OPTICAL CAVITIES AND WAVEGUIDES 3.3 Optical Waveguides 3.3.1 Waveguide Eigenvalue Equation Similar to the cavity case, the Maxwell equations (A.1)-(A.4) are the starting point for the analysis. The expansion ,i(iz-ut) (3.32) is applied to account for both the periodicity in time and in the lon¬ gitudinal direction. The resulting waveguide equation is (3.33) In contrast to the cavity case in Sec. 3.2.2, the angular frequency u can now be seen as an input parameter while the propagation constant 7 is the eigenvalue, er{x,y), uj E(x,y), 7 Waveguide Eqn. (3.34) The solution of a 2D Fabry-Perot (FP) laser problem conforms to formulation (3.34). According to (2.55), the maximum in the gain curve determines the lasing frequency. Note that the choice of input and output parameters in the func¬ tional view of the equation is arbitrary. The frequency u and the propagation constant 7 are dependent on each other by the disper¬ sion relation. As an alternative to (3.34) it is possible to choose £r(x,y), 7 E(x,y), u Waveguide Eqn. (3.35) Usually, there is a natural choice between the two formulations, as for example in the Fabry-Perot case, (3.34) is suitable. In contrast to this stands the solution of a 2D distributed feedback (DFB) laser 3.3. OPTICAL WAVEGUIDES 55 Structure A Structure B Figure 3.4: Slab waveguide. Left: Definition of the structure and the coordinate system. Right: Two possible slab waveguides. Dark shading is equivalent to a high refractive index. For infinite structures (d —> oo), Structure B is expected to leak into the lower region (similar to a high-index substrate). problem. In the 2D problem, an infinite extension of the structure in the ^-direction is assumed, containing a corrugation with period A. The propagation constant is determined by 7 = 1/A. In the actual numerical solution of (3.33) some of the methods conform to the formulation (3.34) while others are of type (3.35). Provided that a good starting guess is known (which is typically the case in optoelectronics) one is free to choose a method of either type and then iterate it until the final solution is found. In the present implementation of the laser simulator, (3.34) is chosen. 3.3.2 Slab Waveguides The slab waveguide is the analogue of the ID cavity. The coordinate system is displayed in Figure 3.4. In the layered media case, the elec¬ tromagnetic fields are decoupled into two separate scalar equations, and the results are the transverse electric (TE) and the transverse magnetic (TM) modes, respectively [32]. For piece wise homogeneous layers, it is possible to solve an ar¬ bitrary slab configuration with a transfer-matrix method (TMM) as 56 CHAPTER 3. OPTICAL CAVITIES AND WAVEGUIDES Coarse Mode Spacing in a: Dense Mode Spacing / Branch Cut Leaky Mode Guided Modes 10 20 lm(Y) [1e15 1/m] Figure 3.5: Waveguide spectrum of a slab structure. The absolute value of a Green's function is plotted on the plane of the complex propagation constant 7. The waveguide structure consists of a guiding layer in vacuum (Figure 3.4, Structure A). The distance d to the artificial boundary is varied in the structure. Top: d = 2 /im. Bottom: d = 8 p.m.. presented in Sec. 4.3.2 and in detail in [83]. The solution can be ob¬ tained for any required accuracy and therefore, the TMM solution is suited for the validation of a finite-element solver. Furthermore, the spectral characteristics of general waveguides as well as the variety of distinct type of eigenmodes can be investigated by the solution of this special structure, as presented in the next section. 3.3.3 Waveguide Spectrum Similar to Sec. 3.2.5, the spectrum of the waveguide modes can be defined as the set of poles that can be observed in the Green's function Ç(r,r',j). Note that the poles remain the same for for all r and r'. In contrast to the cavity case, for waveguides there is a continuous 3.3. OPTICAL WAVEGUIDES 57 part of the spectrum. Discrete vs. Continuous Spectrum The transition from a discrete to a continuous spectrum is illustrated by a layered media slab waveguide that is enclosed by reflecting bound¬ ary condition. Structure A from Figure 3.4 serves as a concrete ex¬ ample. The layer thickness of the slab waveguide is a = 3 /im, the complex refractive index is n = 3 — iO.l, and the wavelength of the light is A = 1 jLtm. The air cladding (n = 1) has a thickness d, for which two values are considered, d = 2 /im and d = 8 p,m. For layered media, Q(r,r',j) can be calculated by a plane wave expansion [84], and the spectrum of the waveguide is presented in Figure 3.5. Guided vs. Radiation Modes With reflecting boundaries at a finite distance d, no energy can es¬ cape to infinity and the spectrum is discrete. The distance d between the dielectric slab and the reflecting boundaries is then gradually in¬ creased. The resulting modes can be divided into two classes: The guided modes that are supported by the dielectric slab. They remain practically the same when d is increased. The other class of modes is supported by the boundary. Further increasing the distance d results in a spectrum with denser poles (compare top and bottom picture in Figure 3.5). In the limit of the boundary conditions at infinity d —Y oo the poles are infinitely near to each other and form a so-called branch cut. The branch cut is the continuous part of the spectrum and the related modes are the radiation modes. Guided and radiation modes form a complete eigenmode basis for a waveguide equation. Propagating vs. Evanescent Modes For small material losses, the branch cut in the 7-plane tightly follows the imaginary axis and a part of the real axis. This gives rise to a further classification of the radiation modes. Near the imaginary axis (Re (7) Leaky Modes Figure 3.5 shows that the dense poles are not independent of each other. The plotted Green's function |£/(7)| encounters a resonance due to the combined influence the dense poles. The variable |(?(7)| has a discontinuity along the branch cut and the resonance only appears on one side of the branch cut. The resonance can be interpreted as a leaky mode. In a practical situation, this is a waveguide mode below cutoff. This mode would be guided by a slightly thicker slab or a slightly higher refractive index. Similar to the guided modes, the leaky mode has a complex wave number 7/. In the spectrum, the pole 7/ is on the other side of the branch cut, mimicking the influence of all dense poles nearby. In consequence, a large part of the continuous spectrum can be replaced by introducing a leaky mode [86]. However, leaky modes are not proper waveguide modes. The optical fields associated with them diverge at least at a part of the boundary, i.e. they cannot be normalized properly [83]. Mathematical Explanation A mathematical explanation of the branch cut is given in the following. The wave vector k of the plane wave expansion can be separated into longitudinal (z) and perpendicular (x) direction, k = yez + kxex. (3.36) The waves propagate in the z-direction and a standing wave pattern is observed in the ce-direction. The spectral portrait is drawn in the complex 7-plane, 7 = V\k\2-k2x. (3.37) Squaring a complex number can be seen as 'folding' a complex plane: two different complex numbers a\ ^ 0,2 possibly map to each other by the square operation, a2 = a2,. In complex analysis, this situation is represented by so-called Riemann sheets [87]. In (3.37), this ambiguity is present because of the term k2. How¬ ever, a choice can be made according to the properties of the standing waves in ^-direction: for Im (kx) > 0, the fields diverge exponentially 3.4. MATHEMATICAL BACKGROUND 59 for |cc| —> oo, while for Im(kx) < 0, proper solutions are obtained. By the J~. operation in (3.37), this choice translates to the branch cut displayed in Figure 3.5 [83]. Consequences for Optoelectronic Device Simulation In summary, it must be recognized that even if the Sommerfeld ra¬ diation conditions are applied correctly, the spectrum has continuous branches. This is characteristic for the waveguide eigenvalue equation. The cavity eigenvalue equation does not show this behavior [83], [88]. The numerical solution by a finite-element approach requires the truncation of the computational domain. Absorbing boundaries such as the perfectly matched layer (PML) material provide an efficient way to include radiation loss in waveguides [89, 90]. An example is presented in Sec. 5.2.1. However, since the diverging modes are not supported, leaky modes replace the radiation modes in the mode spectrum [71]. The divergence property of the leaky modes at infinity does not cause any problems since the numerical solution is found on a finite domain. The approximative replacement of radiation modes by leaky modes remains valid if the absorbing boundaries are placed at a distance d ^> A from the waveguide structure [91]. 3.4 Mathematical Background 3.4.1 Eigenvalues and Eigenvectors The time-independent wave equation is equivalent to the generalized eigenvalue problem Ax = ÇBx, (3.38) with the operators A and B, the eigenvector (or equivalently, eigen¬ mode) x and the eigenvalue £. For optical cavities, the eigenvector can be identified to be the optical field pattern. The eigenvalues are complex quantities. Real and imaginary part can be interpreted as the cavity resonance frequency and the decay constant, respectively. According to the inherent symmetry in the Maxwell equations, A and B are symmetric. However, if lossy materials or radiation to in- 60 CHAPTER 3. OPTICAL CAVITIES AND WAVEGUIDES finity are introduced, A and B become complex. Since they are struc¬ turally symmetric (i.e. symmetric positions in the matrix are equal and not the complex conjugate of each other), the operators become non-Hermitian, and as a consequence, the eigenvalues are complex.1 In the analysis of edge-emitting lasers, the 3D cavity problem can often be separated into a 2D waveguide problem and a ID relation in propagation direction. The 2D waveguide problem defined on a cross-section of the device is also of the structure (3.38). Again, the eigenmodes are equivalent to the optical fields. The meaning of the eigenvalue is different to the 3D problem: it is the complex propaga¬ tion constant, and can be interpreted as the effective refractive index and the decay in the propagation direction. For the solution of the ID longitudinal problem, usually it is not necessary to solve an algebraic eigenvalue problem. The resonance frequency and the decay constant can be found by a root search in the complex plane as presented in Sec. 3.2.4. Together with the result from the 2D waveguide problem, the 3D fields can be obtained. In all cases, the imaginary part of the eigenvalue is related to the optical loss. For an optical solver in the framework of a laser simulator, it is important to calculate the decay of the optical mode since this determines the cavity loss and hence key device parameters such as the laser threshold Ith- As a consequence, the physics has to be formulated for open domains in order to allow radiation to infinity. For analytical calculations, the powerful mathematical theory of complex analysis can be applied. One result is the interdependence of the real and imaginary part of the dielectric function e(r,uj), namely the Kramers-Kronig relations that are summarized in Appendix B. 3.4.2 Quasi-Normal Modes In the derivation of the photon rate equation, the optical field is expanded into eigenmodes, with the resulting eigenvalue equations (3.10) or (2.29). These equations can be found from the Maxwell equations by a formal substitution dt —> —iuj. Equivalent to this sub¬ stitution, the Fourier transform is applied to the differential equations. lrThis is in contrast to the solution of Schrödinger's Equation, where the eigen¬ value problems are complex, but Hermitian. As a consequence, the eigenvalues of the quantum-mechanical systems are real, and can be interpreted as energies. 3.4. MATHEMATICAL BACKGROUND 61 However, for open cavities, the complex Fourier transform has to be applied. While for closed cavities, the field solution can be found from the normal modes, the open cavity problem is expanded into quasi-normal modes. Open Cavities vs. Closed Cavities For closed cavities with no absorption, the eigenvalue equations are real. The system is symmetric under time-reversal t —> — t, so that the characteristic frequency is real. The resulting operators are Hermitian (self-adjoint). Because of the formal similarity to quantum mechanics, there are a wide variety of mathematical tools to solve this problem. The theory becomes non-trivial because optical cavities are usually dissipative. In physical terms, this means that energy can escape and the system is non-conservative. In mathematical terms, the operators that appear become non-Hermitian. From the difference between open and closed cavities, the defini¬ tion of the quasi-normal modes can be stated. First, define the nor¬ mal modes as the solution of the time-independent equations with the boundary condition (3.14) and (3.15), or for a scalar problem, with Dirichlet boundary condition. In contrast to this, the quasi-normal modes are the solution of the time-independent equations with pure outgoing wave (Sommerfeld) boundary condition (3.18). Modes of the Universe vs. Modes of the Cavity In order to prevent the problems introduced by the energy leakage of open cavities, the following idea can be followed: In addition to the finite region of the optical cavity C, an external bath B is defined, resulting in a non-dissipative universe C + B. The resulting basis functions include the modes with high field intensity inside of the cavity as well as all waves outside of the cavity. This basis is often referred to as the 'modes of the universe'. The spectrum contains both discrete and continuous parts. This point of view is always possible and strictly accurate. However, as observed in the discussion of the plane waves (Sec. 3.2.2), the optical modes sometimes contain infinite energy and cannot be normalized. For the systems of a finite extension, completeness and uniqueness 62 CHAPTER 3. OPTICAL CAVITIES AND WAVEGUIDES is guaranteed by the modal orthogonality (2.28). For an infinite ex¬ tension, namely for the 'modes of the universe', it is more difficult to prove the uniqueness since two modes that look almost the same in the vicinity of the cavity may differ at a far distance. This becomes evident for the continuous part of the spectrum: an infinitely small change in the frequency causes a phase mismatch of the waves far away from the scatterer, and therefore, the modes are distinct. As a conclusion, in any situation where the optical modes have to be calculated, one is primarily interested in the discrete confined modes, namely the 'modes of the cavity'. The construction above only solves the problem of energy leakage due to outgoing waves and does not provide a solution for the dissipation due to lossy materials. In a calculation of the 'modes of the cavity', absorption and leakage are responsible for energy loss from the system C. In particular, it can be shown that the discrete spectrum resulting from this solution is sufficient for completely characterizing non-Hermitian eigenvalue problems [92]. Material Loss vs. Radiation Loss Even for closed cavities, it is possible that the system is dissipative. Material loss can be introduced by allowing the permittivity to become complex. In this work, it is proposed to solve optical cavities and waveguides with the means of a finite-element discretization and the solution of the resulting algebraic eigenvalue problem. For such methods, it is important to restrict the problem to a finite computational window. To do this, some artificial materials are chosen in a way that outgoing waves are ideally absorbed. The so-called perfectly matched layers (PMLs) absorb the energy that is actually radiating to infinity. In this case the radiation loss is simulated by material loss in some defined layers surrounding the structure [81]. 3.5 Discussion It has been shown how the cavity and the waveguide situation can be solved for the simulation of semiconductor lasers. The main point 3.5. DISCUSSION 63 is to understand the optical part of the laser diode simulation as an eigenvalue problem. The eigenvector is equivalent to the mode pat¬ tern of the optical field while the corresponding eigenvalue is either the eigenfrequency of the cavity or the propagation constant in the waveguide case. For both cavities and waveguides, the traditional analysis is com¬ plicated by the fact that cavities and waveguides can have inherent losses. For the accurate simulation of laser devices, it is necessary to calculate the cavity and waveguide losses since critical parameters such as the threshold current depend on it. Another optimization requirement is the suppression of the higher order modes. In other words, the losses for the unwanted modes have to be increased in comparison to the losses for the fundamental mode. The mathematical background of the eigenvalue theory for open cavities has been established only recently [33, 93]. In optoelectron¬ ics, the cavities always have losses, which results in non-Hermitian eigenvalue problems and complex eigenvalues. Furthermore, even for high-Q resonators (i.e. cavities with very low losses), it is typically the losses that are of interest to the device designer because of the direct relationship between the cavity loss and the threshold current. In this work, the validation of the simulation code has been done by benchmark devices found in the literature [94]. There are plenty of examples for which analytic solutions exist, as for example with the dielectric slab [95] or the dielectric sphere [96]. However, the ma¬ terial parameters such as the local refractive index or the absorption requires a thorough calibration procedure, as described in Sec. 6.2. The dependencies on the carrier density and the temperature are of particular interest, and the dispersion characteristics (dependence on the frequency) have to be investigated. Chapter 4 Implementation 4.1 Introduction The central goal of this thesis and the work of the Optoelectronics Modeling Group is to provide a simulation tool for laser simulation. It is aimed that the software can be used by industrial device designers, and therefore, the simulator has to be robust, efficient, and easy to use. In consequence, implementation issues become important. This work has been driven by the questions arising from the de¬ vice point of view. Several different numerical schemes have been implemented to answer these questions. They are presented in this chapter. The laser simulations rely on a broad range of physical models. The governing equations have been derived in the previous chapters and are analyzed from a mathematical point of view in Sec. 4.2. For the calculation of the optical wave propagation in arbitrary, passive 3D structures, the Finite-Difference Time-Domain (FDTD) method has been employed. A brief review of the FDTD method is given in Sec. 4.3.1. As a reference, the simple and efficient transfer-matrix method (TMM) has been implemented. It is suitable for ID problems, where there are only forward and backward propagating waves. A summary of the TMM is given in Sec. 4.3.2. Opposed to the traveling- wave optics are the optical eigenmodes. A hierarchy of approximations 65 66 CHAPTER 4. IMPLEMENTATION for the waveguide eigenmode calculation is discussed in Sec.4.4. The finite-element (FE) discretization that has been chosen for the coupled opto-electro-thermal laser simulation is presented in more detail in Sec. 4.5. Finally, implementation details about the eigenmode laser equations are presented in Sec. 4.6, with an emphasis on the self- consistent coupling to the electro-thermal solver. 4.2 Analysis of the Governing Equations The relevant differential equations are Laplace equation: d2(j) + d2 Rate equation: 1/r • S — dtS = 0 (4-2) Diffusion equation: D d2W - dtW = 0 (4.3) Wave equation: v2 • d2xE - d2E = 0, (4.4) with the time constant r in [s], the diffusion constant D in [s/ra2] and the propagation velocity v in [m/s]. Note that the formulation of the non-equilibrium carrier dynamics in the density matrix for¬ mulation also leads to equations of the form (4.2) for each matrix element. Together with the intra-band scattering terms, this results in a diffusion-type equation in k-space [39]. While (4.2) and (4.3) are parabolic equations, the Laplace equation (4.1) is elliptic and the wave equation (4.4) is of the hyperbolic type [97]. This has some consequences on the numerical solution as well as the physical meaning of the partial differential equations. The Laplace equation has no time dependence and therefore does not describe any energy exchange at all. While in the parabolic case, an initial field distribution smooths out irreversibly, the hyperbolic equation lets it travel at a finite velocity and scattering at the inhomogeneities of the structure. Furthermore, the boundaries have to be treated differently. While Dirichlet or Neumann boundary conditions (see Sec. 3.2.2) are suitable for parabolic and elliptic equations, absorbing boundaries are needed for the wave equation. Transforming the hyperbolic equation to the frequency-domain seemingly solves the problem with traveling waves, but introduces a strong non-local character of the problem 4.3. TRAVELING-WAVE OPTICS 67 (eigenmodes). A small perturbation of the structure may change the resulting field completely, possibly with no resemblance to the original field pattern. In general, a similar perturbation of a parabolic problem will only affect the near surroundings. 4.3 Traveling-Wave Optics 4.3.1 Finite-Difference Time-Domain Method Maxwell's curl equations (A.9)-(A.12) can be discretized in time- domain by the Yee algorithm [79]. The fields are discretized in space on a staggered grid. This allows to replace the curl operator with central differences. Applying central differences to calculate the time evolution results in the so-called leapfrog scheme. As an example, the z-component of (A.9) is dEy ÔEX ÔHZ = -d~-dir "M^T' (45) and in its discretized version reads - Et Et - (x+Ax/2,y,z) (x-Ax/2,y,z) E\. (x,y+Ay/2,z) El (x,y-Ay/2,z) Ax Ay rrt+At/2 ut-At/2 nz — nz (x,y,z) {x,y,z) / A n\ "" ' (46) M The equation can be solved explicitly for Hz (x,y,z), requiring only the components of earlier times. Similar expressions can be found for the electric field Et+At. All fields can be solved by marching on in time, and no large system matrix has to be inverted. Even though first-order differences are applied, second order accuracy can be achieved [98]. The stability of the algorithm can only be assured if a small enough time step is chosen according to < (4.7) At min —, K ' c^/l/(AxY + l/(AyY + l/(AzY Furthermore, the spatial discretization has to be fine enough to re¬ solve the optical wave. Ax œ A/20 is enough for general structures. 68 CHAPTER 4. IMPLEMENTATION However, it turns out that the numerical errors in layered media sum ~ up, and for the modeling of highly reflective Bragg mirrors Ax A/40 is recommended in the direction perpendicular to the layers. A phase correction scheme to overcome this restriction is presented in [99]. For the domain truncation, Perfectly matched layer (PML) bound¬ aries are applied to model open-domain problems [79]. In order to solve problems with a rotationally symmetric structure, the Yee algo¬ rithm has been implemented in the Body of Revolution (BOR) sym¬ metry [76]. The FDTD method is a rigorous method suitable for treating op¬ toelectronic device structures [100]. It is capable of treating big com¬ putational domains with a modest amount of memory because the algorithm does not require the inversion of the system matrix. The FDTD method has been applied to the calculation of spontaneous emission in a dielectric microdisc [101], in photonic crystals [102, 103] and in vertical-cavity surface-emitting lasers [104]. 4.3.2 Transfer-Matrix Method In the special case of a ID structure, the wave propagation is restricted to right and left traveling waves. Having explicit expressions for the waves, a transfer-matrix formalism can be employed to solve the open cavity problem. From Sec. 3.2.4, the inhomogeneous wave equation for ID struc¬ tures is d2zE(z) + ^E(z)=F(z). (4.8) The waves at two points i and i + 1 in the structure can be related by (4M> { E~(Zi) )-\T21 T22 ).{ E~(zi+l) ) With a ID discretization of the cavity, neighboring points can be con¬ nected with the above equation. The advantage of the TMM is that subsequent segments can be taken into account by a simple multipli¬ cation of the respective transfer-matrices. 4.3. TRAVELING-WAVE OPTICS 69 The transfer-matrices for an index step n\ —> n2 and for a homo¬ geneous section of length d are pikd 1-^2 ) , (4-10) Td=[ 0 e-ikd with n!-n2 2y/n1n2 . . = = = = and • r\i -r2\ ; t12 t2\ ; (4.11) n\ +n2 n\ + n2 For a periodic corrugation, er(z) depends on z with the given period A. For certain corrugation types, closed solutions for the wave equation exist [1]. With these expressions, a matrix formalism can be derived for structures with piece wise periodic corrugations or even some types of chirp corrugations [9]. However, it is also possible to allow an arbitrary variation in £r(z) and to solve the longitudinal problem with a small longitudinal spac¬ ing. As soon as the temperature and carrier density dependence is included, the regularity necessary for the closed expressions is lost and the more rigorous solution is required. The TMM is directly applicable when a source is present in the structure. For the application in optoelectronics, this is needed to calculate the reflection coefficient of layered media as presented in Figure 3.3, Sec. 3.2.4. However, it is also possible to solve for the eigenmodes of an optical cavity, which is presented in the following. From (4.8), Q(z, z>)F(z)dz> _ £ E(z) ~ ( ] —WjMzfl—' with the Green's function Q(z,z') and the Wronskian W[£r(;z)]. For proper definition of W and Q, see [24, 105]. If the eigenvalue solution is sought, the source term vanishes, F(z) = 0. It is evident from (4.12) that in this case, there can only be a nonzero solution if W[er(z)] =0. (4.13) 70 CHAPTER 4. IMPLEMENTATION Note that W is a functional of er(z). In a laser, sr(z) is dependent on the carrier density and temperature. The requirement (4.13) is the global condition that has to be fulfilled for a stationary solution. An intuitive interpretation related to the application in laser sim¬ ulations observes the electromagnetic waves propagating in a laser cavity. The wave is amplified in the active region and reduced in am¬ plitude by the reflection at the semi-transparent mirrors. The gain and phase shift of the wave can expressed by the complex variable grt. For the cavity eigenmodes, the steady state condition requires that the round-trip gain grt has to be unity. Amplitude \grt\ = 1 and phase Lgrt = 0 have to match for any location in the cavity. The round-trip gain depends on two free parameters, and therefore, the two equations can be satisfied. One parameter is the optical fre¬ quency uj. The second parameter, a, can be interpreted as optical gain (for cavities with an active region) or the cavity decay constant (for passive optical cavities). Since grt is an analytic function, it can be shown that from all combinations of the two free parameters, a pair (uj, a) can be found for which grt = 1. In general, it does not matter at which position inside of the cavity the round-trip gain is calculated. However, the numerical stability of the algorithm can be enhanced when the round trip gain is not evaluated at a position with vanishing field amplitude. For example, when calculating VCSEL eigenmodes, it is better to chose some point in the centre and not at the end of the Bragg mirrors. 4.4 Hierarchy of Approximations for Waveguide Solutions Next, the solution of the transverse eigenvalue problem is discussed. It is often possible to simplify the eigenvalue problem by an ap¬ proximation to the full-wave treatment. Comparing an approximate eigenmode analysis with a rigorous one, there is not necessarily an obvious correlation between the resulting modes. The approximation introduces a mode degeneracy that is removed in the rigorous con¬ sideration. In Sec. 3.2.3, the designation of VCSEL modes has been made using the nomenclature used for circular waveguides. The prob- 4.4. WAVEGUIDE APPROXIMATIONS 71 lern has to be re-visited here for the analysis of a general waveguide cross-section. With the use of the normalized frequency V and the characteristic variation of the refractive index -#, V = /crov^i - £o (4.14) 0 = — - 1, (4.15) so the approximations made by the simplified models can be quantified. For simplicity, it is assumed that the dielectric profile has a step- index profile, with a piece wise constant permittivity s(x,y) = £q in the cladding and e(x,y) = £\ in the core. However, the methods are not restricted to piece wise constant materials. The parameter ro is the characteristic dimension of the core, for instance the ridge width in EELs. Weak Guidance Approximation Starting from (3.33), the vectorial fields can be split into transverse and longitudinal components. For the electric field, E = (Ex, Ey,Ez)T and Et — (Ex,Ey)T. By the use of vector calculus (see e.g. [32]), = • ( . AT + ^-Sr -j2JEt Vt (erET VT(s~1)) (4.16) The right hand side of (4.16) is negligible for structures with low dielectric contrast and large near fields, i.e. ê —> 0 (while V remains finite) [71]. The differentials of the dielectric function er(x,y) must be understood in terms of distributions. The right hand side of (4.16) can be expanded in powers of ê, deriving a general theory of higher order corrections [71]. In first order approximation, (4.16) reduces to the scalar equation. The weak guidance theory is a rigorous formalism in the way that an arbitrary order of accuracy can be obtained just by choosing the power of i9 in the series. The requirement of a low dielectric contrast is violated for many applications in integrated optoelectronics. However, as a special case of the weak guidance approximation, a waveguide embedded in a low 72 CHAPTER 4. IMPLEMENTATION index medium (such as a ridge waveguide in air) can be approximated by imposing Dirichlet boundary conditions at the material boundary between semiconductor and air, and applying the weak guidance ap¬ proximation to the remaining waveguide [71]. The domain truncation at the semiconductor-air interface is arbitrary, and the accuracy of the approximation cannot be quantified because it depends on the structure under investigation. Semi-Vectorial Approximation In contrast to the $n-series of the weak guidance theory, the semi- vectorial approximation has an accuracy that is intrinsically limited. This method completely neglects the minor component of the trans¬ verse field, thus reducing ET = (EX,EZ)T to either Ex or Ey. [71]. As a result, one gets modes with Ex and Hy as main component, respectively. In principle this is a suitable approximation for laser simulations because it distinguishes between the two possible polar¬ ization modes. However, there is no estimate for the approximation error. Also, the savings in comparison to the rigorous treatment are rather in implementation effort than in computational resources. Requirements for Laser Simulation In the analysis of integrated optics, it is common to use one of the several approximations to the waveguide problem. However, when used in conjunction with a laser simulator, the approximations to the waveguide problem have the disadvantage that they fail to predict the waveguide loss accurately, in particular, for structures where ra¬ diation modes are present. As a conclusion, the waveguide problem has to be solved rigorously, and proper absorbing boundaries has to be employed. Concerning the CPU time for a numerical scheme, it is more advisable to optimize the re-solving of the algebraic eigenvalue problem, instead of applying an approximative waveguide problem. An example of the rigorous solution is given in Sec. 5.2.1. 4.5. NUMERICAL EIGENMODE SOLUTION 73 4.5 Numerical Eigenmode Solution For the discretization of the general wave equation (3.10) and (3.33), a finite-element method is employed, as presented in detail in [81]. The introduction of the BOR expansion (see Sec. 3.2.3) allows to treat circular-symmetric structures. Following the general procedure of the finite-element method, the differential equations are written as a variational functional and the edge-element basis functions presented in [106, 80] are applied. Next, the formula is differentiated in order to find the stationary point of the functional. As a result, a sequence of algebraic sparse eigenvalue problems is obtained. Similar to their continuous counterparts, they are generalized, complex, symmetric and non-Hermitian. Due to the choice of linear edge-elements, they have the following structure, ATT ATn Un 1 UJ. Btt 0 if>r AnT Ann ^ 0 Br u I J „ (4.17) with the vector tj) containing first the coefficients that determine the transverse .E-field component (T-subscript) and then the coefficients for the normal E-field component (n-subscript). The mode index v stems from the BOR expansion (3.23). Note that the 2D edge-element basis functions consist of edge- elements for transverse components and nodal elements for normal component. Consequently, the vector length in (4.17) is equal to the number of nodes plus the number of edges in the mesh. Higher order elements could be employed in the same manner, and an evaluation of the overall efficiency is left for future investigations. Furthermore, PML absorbing boundaries are used. Consequently, a general anisotropic media have to be considered, and as a result, the materials become complex diagonal tensors. Finite-elements of the tensor are described in [90] and, for the BOR case, in [107]. For each v in (4.17), an algebraic eigenvalue problem (3.38) has to be solved. The JTacobi-Davidson QZ Iteration Method is employed to do this. Details are described in [81, 108]. An equivalent formulation of the eigenvalue problem can be ob- 74 CHAPTER 4. IMPLEMENTATION tained from the characteristic polynomial det([A]-^[B]) = 0. (4.18) The effort to find a solution of (4.18) is drastically reduced if a good starting guess for £ is known. The speed of convergence of the iterative method can be improved by using a preconditioner, that is, an approximate solution of the eigenproblem that is used in the algorithm in a correction step. As a preconditioner, [K] = ([A]-mr1 (4.19) is solved, where £ is a guess for the eigenvalue uj2/c2. Another possible choice is the inversion of the trace of the above equations, which can be calculated without a matrix inversion [108] but with the expense of a bad convergence of the iterative method. The optimum choice of the preconditioner is still a research topic and at the moment, no alternative to the computationally expensive equation (4.19) is available. When used in conjunction with the laser equations, (4.19) can be solved once at the beginning of the simulation, and can be used to enhance the re-iteration of the optical problem as described in Sec. 4.6. As an additional step in Figure 4.2, the calculation of the preconditioner can be done in advance, outside of the loop. 4.6 Eigenmode Laser Equations The eigenmode laser model has been derived in Sec. 2.4 and a complete set of the semiconductor equations can be found in Appendix C. The numerical solution of the coupled electro-thermal equations has been done in [20, 40, 109]. From the implementation point of view, it is important to split the optical treatment into the calculation of the optical mode pattern and the photon rate equation. Note that the mode pattern consists the normalized optical fields ^, while the magnitude of the fields is contained in the photon number S. In physics-based device simulation, it is advantageous to formulate the physical models independently of the device dimensions. It should 4.6. EIGENMODE LASER EQUATIONS 75 be possible to perform ID, 2D and 3D simulations without changing anything except the geometry description. This has been successfully formulated for the electronic equations [110, 111]. In contrast to the electronic relations, the optical eigenvalue problem is non-local. Fur¬ thermore, the different coordinate directions usually cannot be treated equally. As e.g. in the case of EELs, a 3D structure can be separated into simpler physical models of lower dimensionality. It is a challenge to allow a flexible treatment of the optical modes and at the same time keep the interface to the device simulator simple. From Cavity Eigenmodes to Waveguide Eigenmodes EELs have a longitudinal extent typically between 0.5 mm and 1 mm, while the transverse dimension is limited to a few microns. With this extreme aspect ratio, it is possible to separate the transverse and the longitudinal optical field. In a 2D simulation, the waveguide eigenvalue problem (2.54) has to be solved. The complex propagation constant 7^ — 7^ + i 7" provides the effective refractive index 'seen' by the wave, neff^ = cyl/uj, and the loss due to both absorption and radiation, nv = cy"/uj. Note that the mirror reflectivities may be different for the distinct modes v due to the different modal propagation constants or due to the scattering at the facets. In all cases, not only the optical field but also the other quantities such as the carrier densities have spatial inhomogeneities. Further¬ more, if temperature is taken into account, the heat flow is not purely transverse but shows a different behavior at the mirror facets. As a consequence, longitudinal current flow has to be expected and a full 3D simulation has to be considered. Note that the optical problem can still be separated into (possibly more than one) transverse waveguide problem and the longitudinal ID cavity problem. Figure 2.4 illus¬ trates how the transverse and longitudinal results can be combined to get the 3D optical field pattern. If the longitudinal current flow shall be included, the electro-thermal problem has to be solved in 3D [112]. CHAPTER 4. IMPLEMENTATION Device Structure Optical Eigenmodes Lasing Frequency Newton Figure 4.1: Simple simulation flow. The Poisson equation, the continuity equations and the photon rate equations are solved self- consistently by the Newton scheme. In a this simulation mode, the optical mode pattern is solved prior to the ramping of the bias. How¬ ever, the lasing frequency is updated during the simulation run. Figure 4.2: Simulation flow for optical re-solving. The Helmholtz equation is only weakly coupled and can be updated outside the New¬ ton iteration in a decoupled iteration. After convergence of this in¬ termediate loop, the device is re-calculated in the outer loop for the next bias point. 4.6. EIGENMODE LASER EQUATIONS 77 Self-Consistent Coupling Figure 4.1 summarizes the simulation flow [113]. Prior to the self- consistent calculation of the electronic equations and the photon rate equation, the spatial optical distribution and the lasing wavelength are determined. Re-iteration is done until global convergence is achieved. For an index-guided Fabry-Perot laser, only few iterations are nec¬ essary because the remaining dependencies between optical and elec¬ tronic part are weak [41]. The nonlinear system of electronic equations is solved by the New¬ ton method using a box discretization. Due to the eigenvalue nature of the optical equations, it is not possible to include them into a standard Newton updating scheme. However, only the photon number S and the cavity gain G and loss L are strongly coupled to the electronics, while the optical field distribution is weakly coupled. As an approximation to (2.35), the gain is calculated from an overlap integral of the local material gain Rst(r), Gv = (J Rst(r)\$„(r)\2d2r, (4.20) while the loss Lv is given by (2.57). The actual lasing frequency is also derived from the longitudinal cavity solution. The strong coupling between electronics and optics can be ex¬ pressed solely by the photon rate equation (2.64) which is fully in¬ cluded in the Newton updating scheme and solved self-consistently with the electronic equations. The problem of singularity of the pho¬ ton rate equation at lasing threshold is overcome by using an ad¬ ditional slack-equation which provides quadratic convergence of the Newton algorithm even at threshold [114]. Figure 4.2 contains an improved simulation flow that considers a repeated solution of the optical field. Due to the carrier-induced change in refractive index, the optical field pattern can change con¬ siderably when ramping the diode laser [43, 115]. The optical cavity problem cannot be included in the iterative Newton scheme because of the non-local nature of the optical eigenmodes. However, due to the weak coupling between the optical mode and the electronics, it is feasible to do a Gummel-type outer iteration. A robust and efficient scheme can be found with optimizing the convergence criterion. 78 CHAPTER 4. IMPLEMENTATION 4.7 Discussion In summary, contributions have been made to several numerical meth¬ ods, while keeping a focus on the application of the software in op¬ toelectronic device design. The different models have been evaluated, resulting in the decision that the eigenmode laser equations are best suited for industrial design of laser devices. Compatibility to Existing Software A major goal was to build a computer program that is robust and easy to use. The commercial TCAD suite of ISE Integrated Systems Engineering AG has been employed for the peripheral tasks such as structure generation, grid generation, visualization, and parameteriza¬ tion. The compatibility with a professional software environment is essential for the handling of complex devices and the physical models needed in optoelectronics. From the user's perspective, all laser models are incorporated into the device simulator DESSIS-ISE. Consequently, the development of the laser simulator had to be done inside of the big software project at ISE AG, and software engineering issues have become relevant. Regarding the solution of the optical modes, it was possible to define an interface to the electro-thermal simulator. The eigenmode solver is compiled in a separate library, LUMI, which proved to be useful for both code development and testing. System Requirements Rigorous device simulation has one drawback: it needs a lot of compu¬ tational resources. While the empirical rate equations can be solved within milliseconds, a device simulation can become CPU intensive. However, today's development in TCAD software has to be designed for tomorrow's computer architectures. While current personal com¬ puters typically have a single processor and 500 MB of main memory, a full-featured simulation task may be distributed on several CPU's and can occupy 2-8 GB of main memory. However, when TCAD is employed in conjunction with the processing of prototypes and their 4.7. DISCUSSION 79 characterization, one has to see computing costs in relation to mea¬ surement equipment and time savings in the process cycle. For both the memory and the CPU requirement, it is important to find a balance between all the implemented models. As an example, it can easily be calculated from the number of vertices in the grid how much memory the electro-thermal simulation will occupy, and it can be seen that the rigorous solution of the optical problem must not be compromised because this would not save much in the overall memory requirement. Furthermore, the hierarchical modeling approach and its support by the software environment can be invaluable to the device designer. Hierarchical modeling is typically employed in many simulation as¬ pects, e.g. by the variation of the device dimensionality (ID - 2D - 3D) and the choice of the physical model (isotherm - thermodynamic - hydrodynamic), any degree of complexity can be reached, from run times in seconds to several days [112]. Chapter 5 Results 5.1 Introduction In this chapter, simulation results for the optical behavior of laser diodes are presented. For a physics-based device simulator, it is im¬ portant to cover a wide range of applications. Only when a variety of examples can be solved successfully, it can be shown that the physical models are implemented in a general manner, and that they allow to predict the characteristic of new devices. However, this chapter is not intended to cover the laser feature of DESSIS-ISE systematically. An application note describing the standard functionality of the simulator is available1 and a tutorial including a variety of examples has been provided in [116, 109]. Furthermore, some applications of the laser simulator have been presented in [41, 42, 117, 112, 118]. Simulation results are presented for edge-emitting lasers in Sec. 5.2, as well as for VCSELs in Sec. 5.3. The examples include radia¬ tion leakage into substrate (Sec. 5.2.1), spontaneous emission coupling (Sec. 5.2.2), and the calculation of the 3D optical mode pattern in VC¬ SELs employing the FEM (Sec. 5.3.1) as well as the FDTD method (Sec. 5.3.2). lrThe web site of the Optoelectronics Modeling Group provides updated infor¬ mation about the laser features of DESSIS-ISE, http:\\www.iis.ee.ethz.ch\~laser 81 82 CHAPTERS. RESULTS Figure 5.1: Simulation results: Radiation leakage into substrate. A ridge waveguide grown on a high-index substrate typically shows some waveguide loss due to leakage into substrate. The variation in the spacer thickness affects the substrate loss considerably, while the ef¬ fective refractive index is almost unchanged. 5.2 Edge-Emitting Laser Diodes 5.2.1 Radiation Leakage into Substrate There are three possibilities for optical losses in an edge-emitting laser: the mirror loss, the absorption in the semiconductor material and scattering losses. As shown in Sec. 4.6, absorption loss and radiation loss can be calculated by the solution of the transverse eigenvalue problem. For the device design, absorption and scattering losses must be kept as low as possible. The common InGaAlAs material system, grown on GaAs sub¬ strates, shows the potential for a considerable radiation leakage into substrate. This is due to the higher refractive index of GaAs and con¬ sequently the anti-guiding at the interface to the substrate material. In general, the additional loss mechanism should be avoided. How¬ ever, it has been proposed to employ the substrate leakage to achieve high power single mode lasing [94]. The scattering losses are different for the fundamental and the next higher order modes, and with ac¬ cepting a somewhat higher threshold for the fundamental mode, the losses for the higher order modes can be increased considerably, and a high single-mode lasing output can be achieved. For the solution of the waveguide eigenvalue problem, it is neces- 5.2. EDGE-EMITTING LASER DIODES 83 sary to introduce absorbing boundary conditions (ABC) at the domain truncation in order to model radiation losses [119, 81]. For a gen¬ eral optoelectronics simulation tool these capabilities are essential, even for devices with a substrate loss much smaller than the mirror and absorption losses. Truncating the computational domain with a boundary condition that does not allow electromagnetic radiation can lead to completely artificial mode patterns, even if the correct optical fields vanishes at the boundary. A spurious optical mode (a so-called substrate mode) is then guided by the high-index substrate and the domain truncation. The spurious modes cover the frequency range of interest, and therefore, they cannot be separated from the true waveg¬ uide solution. An optical mode solver equipped with an ABC solves the problem correctly, because the substrate modes are not present. The numerical example presented in Figure 5.1 shows how the substrate leakage can be controlled by the variation of the spacer thickness. Starting with a GaAs substrate (n = 3.65), there is a spacer layer (n = 3.35) and a high refractive-index guiding layer. In a typical laser design, this is often called the separate confinement layer because it allows to confine the charge carriers in the active region separately from the optical mode. The ridge layer is again of a lower refractive index, and the lateral confinement is achieved by the etching of a ridge structure. As can be seen from the resulting eigenvalue, the variation in the spacer thickness affects the substrate loss considerably. 5.2.2 Spontaneous Emission Coupling This section deals with the electromagnetic radiation originating from a spontaneous emission event in an edge-emitting laser. While free space radiation shows a power decay according to 1/r2, the waveg¬ uide can support lossless modes in the direction of the waveguide. In the far field pattern (or its square, the radiated power density per unit angle), the lossless modes appear as Dirac distribution functions. In¬ tegration over these lossless modes results in the guided wave power. In the following, a method is presented to calculate the coupling ratio between the dipole and the eigenmodes of the waveguide. In Figure 5.3, the simulation setup is explained. The refractive index of the substrate is nsubs — 3.1 while the vertical guiding is 84 CHAPTERS. RESULTS a) b) e) f) Figure 5.2: Optical modes in a ridge waveguide, a) - e) shows the field intensity as well as the polarization of the optical mode. The number in the inset is the effective refractive index neff. f) shows the device structure including the discretization for the finite-element calculation. achieved by a semiconductor layer of nguide — 3.5 and thickness d — 95 nm. The wave is confined by a ridge waveguide consisting of a nitride isolation (nnn — 2.0) and a metal cladding (assumed perfectly electric conducting, PEC). The ridge width is 2.6 p,vn at the basis and 1.5 yum at the top. The ridge height is 0.74 ^m. A dipole emitter is placed in the active region in a transverse plane p — (x,y)T at z = 0. The radiation from the dipole source is simulated. The simulated field pattern is shown in a horizontal plane and also in a second plane p' = (x',y') at some distance from the dipole source z' ^> A. In the //-plane, the optical field is expanded into the eigenmodes of the waveguide. If the distance is large in comparison to the wavelength À, only the waveguide modes supported by the structure are present and all other modes have died away. 5.2. EDGE-EMITTING LASER DIODES 85 Figure 5.3: Setup and Results of the FDTD simulations. The struc¬ ture has been illuminated by a narrow-band pulse at a point rsrc. The radiation from the source is shown in the horizontal plane. At some distance d, a field plot is shown in a vertical plane. The first step is to analyze the transverse eigenmodes of the waveg¬ uide. In the structure under investigation, the solution of the eigen¬ value problem reveals that the structure supports five eigenmodes. For a given free space wavelength (Ao = 1 £im), the propagation con¬ stants 7j, are obtained from the solution of the eigenvalue problem. The results are summarized in Figure 5.3. For the calculation of the spontaneous emission coupling the elec¬ tromagnetic field problem has to be solved in three dimensions. A concentrated source (electrical dipole) is placed at one end of the sim¬ ulation domain. Some portion of the emitted power will be scattered into the waveguide modes. This energy is traveling down the waveg¬ uide without any loss or coupling between the modes. At a distance far enough from the emitter (I ^> A), the transverse field can be mea- 86 CHAPTERS. RESULTS 3 CO N CD II >^ o" II X *à*kHàakk, 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time [ps] Figure 5.4: Results of the FDTD simulations. The excitation is placed at the centre of the waveguide. The different curves show the time- domain response of the different sensors placed along the cavity. sured and expanded into the waveguide modes. Dividing the power contained by each of the waveguide modes by the total emitted power, the spontaneous emission factor can be calculated. The numerical method used for the 3D simulation is the finite- difference time-domain (FDTD) method [79]. FDTD results are in¬ herently in the time-domain, while the presented analysis is in the frequency-domain. The Fourier transform is applied to obtain E(uj) — T{E{t)} and H(uj) = F{H(t)}. From the transformed fields the power through a closed area can be obtained, yRAD (uj) = (U)E(uj) xH*(uj)dA. (5.1) In the source location, it is convenient to define the voltage U(t) Ey (t) • Ay and the source current I(t) jyRC(t) • Ax Az. The total power emitted by a dipole can be calculated from U(uj) = T{U(t)}. I(uj)=f{I(t)}: ySRC (uj) = U(uj)-F(uj). (5.2) Note that a transparent source must be chosen ('added source'), scaled such that it acts as a current density jSRC imposed on the simulation domain. On the other hand, the voltage U(t) is measured across the 5.2. EDGE-EMITTING LASER DIODES 87 10 r^" 0 10 20 30 40 50 60 70 80 Distance [|im] Figure 5.5: Results of the FDTD simulations. The relative spectral power density at the different sensor positions is shown in the figure. The data is scaled to the power density of the source. The continu¬ ous line shows the beating oscillation that is met asymptotically with increasing distance from the concentrated source. There are only two modes present in the beating is due to the difference in propagation constant ju. With the knowledge of the waveguide eigenmodes, the spontaneous emission coupling factors can be extracted. same edge, recording both the imposed voltage as well as the E-field scattered by the surroundings. The discrete model of the waveguide structure contained 180 x 200 x 1500 cells with a cell size of 0.013 /im. As a source, a narrow¬ band pulse has been chosen. From the signal response of a single simulation run, information can be obtained over the entire frequency range of interest. The excitation has been chosen at the device cen¬ tre, and in this special case, only two of the transverse eigenmodes (Figure 5.2a and c) can be excited. In order to observe the wave traveling down the waveguide, sensors have been placed with a longi¬ tudinal spacing of 0.75 //.m. A folded cavity has been chosen in order to save computational resources. To achieve this, the cavity has been terminated at the far end by a perfectly conducting plane so that the pulse is reflected and travels back. Due to the finite pulse length, the forward and backward traveling pulses can be separated easily. Results are presented in Figures 5.4 and 5.5. From symmetry 88 CHAPTERS. RESULTS considerations, only two guided modes can be excited with the special placement of the source. Since the propagation constants of the two modes are known from the transverse eigenvalue problem, the optical power in each of the modes can be extracted from the beating observed in the sensors. From the beating oscillations, the portion into the fundamental as well as the fourth mode can be calculated, A)(/>o,^o) = 3.4-nr4 (5.3) /?4(/>o,^o) = 2.6-10-4. (5.4) The calculation has been simplified because of the special place¬ ment of the dipole. If the excitation is not placed on the symmetry axis, the total guided power is split into all the different waveguide modes instead of only two. It is important to see that the amplitudes of the different modes as well as the phase between the modes have to be considered. Due to the different propagation constants jv of the eigenmodes, the phase relations are changing along the propagation direction. In order to obtain amplitudes and phase relations of the modes, the FDTD fields have to be recorded at two cross-sections, separated by an arbitrary distance, av(uj) = IIE(uj,p,z2) • *„ e**» dp (5.5) av(uj) = IIE(uj,p,Zl) • $„ ë+» e*ï"(**-*0 dp, (5.6) where E(uj,p,z) is the Fourier transform of the FDTD field in the cross-section at position z, E(t,p,z). The resulting system of equa¬ tions can be solved to obtain the amplitude as well as the power guided in each mode. = — For numerical reasons it is advisable to choose Az z\ Z2 ~ 7/4, where 7 is the mean value of the propagation constants 7^, v=n—1 7=1/« Yl > (5-7) i/=0 The proposed numerical scheme does not take into account the amplification of the traveling wave in the section between the emission 5.3. SURFACE-EMITTING LASERS 89 and the location where the guided modes are established. However, since the near field of the emitter decays proportionally to the square of the distance, the portion of the radiation that is amplified and then scatters into a guided mode is relatively small. In the present example it can be seen from Figure 5.4 (and Figure 5.5) that after about 30 //.m (or 0.3 ps), the near field of the emitter is much smaller than the guided field. In comparison to the photon life time, this is small enough to be neglected. However, it has been shown that for gain- guiding configurations, the spontaneous emission can considerably be enhanced by this effect [120]. 5.3 Surface-Emitting Lasers The rigorous calculation of the optical field pattern in VCSELs is a challenge because in contrast to the EELs, no separation between the longitudinal and transverse directions can be made. Various methods have been proposed to solve this problem. In the following, results are presented from two approaches that differ in many aspects. 5.3.1 Finite-Element Optical Mode Calculation The solution of the cavity equation (3.10) can be done by first ex¬ panding the fields in the BOR manner as described in Sec. 3.2.3. The discretization has to deal with the vectorial nature of the fields, and therefore the edge-element expansion functions are applied and the algebraic equation (4.17) is obtained. As an example, a device with a buried mirror has been investigated [81]. The structure is a VCSEL containing a dielectric mirror on one side and a semiconductor DBR stack on the other side. The advantage of the dielectric mirror is that a higher refractive index contrast can be obtained and therefore, a few periods are enough to achieve a high reflectance. The structure has been discretized by a finite-element mesh using nei « 115'000 elements, nnd ~ 115'000 nodes and neg œ 230'000 edges. The resulting sparse matrix order can be calculated by the sum n-nd + n>ed ~ 345'000. As described in Sec. 4.5, the pre-conditioner is chosen such that it is necessary to invert the resulting matrix once. 90 CHAPTERS. RESULTS Figure 5.6: Finite-element result of the VCSEL eigenmodes. Left: De¬ vice Structure. Right: Fundamental and higher-order mode. The top Bragg mirror is formed by 62 quarter-wavelength layers of semicon¬ ductor material while the bottom Bragg mirror is a dielectric mirror. Due to the higher refractive index contrast of the dielectric material, a reflectivity of > 95 % can be achieved with a few layers. A direct solver is applied for this task [121]. It solved the problem in five minutes on a Compaq Alpha Server ES40 (667 MHz) allocating 2.3 GBytes of memory. In Figure 5.6, the fundamental mode and one higher order mode are displayed. From the eigenvalues, the resonant wavelengths and the photon life time for the fundamental and the higher order mode can be calculated. 5.3.2 Optical Modes Derived by FDTD As a second method to obtain the optical modes in a VCSEL cavity, a calculation using FDTD is presented [78]. As required by FDTD, the device structure is discretized by a tensor-product mesh. A con- 5.3. SURFACE-EMITTING LASERS 91 Figure 5.7: Steady state mode pattern inside the VCSEL cavity. The real part of the complex eigenvector is shown in the Figure. (1) Fun¬ damental mode. (2) Laser output. (3) Bragg losses. (4) Radiating waves. (5) Guided surface waves. centrated dipole is placed inside of the active region, and the cavity is excited by a band-limited pulse. In principle, the calculated response of the cavity is a superposition of a huge number of eigenmodes. However, the VCSEL modes can be identified by their decay constants. Increasing the injection current, the lasing modes with smallest total losses appear first. Therefore, the eigenmodes are sorted by their decay constants. The fundamental mode is the one with the longest life time and the higher order modes have shorter lifetimes (higher decay constants). The time-domain method with subsequent extraction of the relevant eigenmodes works so well because the laser mirrors are frequency selective. Therefore, the radiating modes couple strongly to the continuum of the free-space modes and decay rapidly with time. As a consequence, the formation of the eigenmodes evolve quickly and the FDTD simulation time is reasonable. A combination of FDTD and the Padé approximation has previ¬ ously been applied to the simulation of cavity resonant frequencies and quality factors [122]. The procedure is as follows. The Fourier trans- 92 CHAPTERS. RESULTS Higher Order Mode Fundamental Mode 1 2 Radial Distance [|im] Figure 5.8: Mode pattern in the active region. The fundamental mode and the next higher mode are shown. The device structure is similar to the one presented in Figure 5.7. The active region is a horizontal plain in the center of the A-cavity. form is applied to the time-domain signal, s(uj) — ^{sty)}. Since s(uj) is known to be an analytic function, it is approximated by a polynomial \N i=0a'< UJ" s(uj) N (5.8) i + £ Li ßi"* By the application of the Fast Fourier Transform (FFT), s(uj) is de¬ fined on discrete sampling and the coefficients ckj and ßi can be deter¬ mined by a singular value decomposition. The roots of the polynomial 1 + X^=1 ßiOJ1 are good approximations of the poles of s(uj). From the real and the imaginary part of the poles, the eigenfrequency and the decay constant can be derived. The Padé approximation is applied individually to the time series taken at the vertices in the active region. If the simulation has evolved long enough for the high-order modes to vanish, the extracted poles from different vertices in the active region converge and can be used to approximate the eigenvalues. In order to illustrate the applicability of the presented method, 5.3. SURFACE-EMITTING LASERS 93 J* tn 1 6um . N 'o 1 6 " 1 4 (/) / £Z 5um , c O i 2 CD 3)im 4 um ^^' Q 2um 1_ CD s: 2 4628 2 4636 2 4644 o Q. Resonance Frequency [10" s"1 o Q. O 0.0 1.2 2.4 3.6 Distance from Center [um] Figure 5.9: Comparison between devices with different aperture diam¬ eters d. The aperture is varied between d — 2 yum and d = 5 /im. The change in the mode pattern, measured in the active region is shown in the figure. In the inset, the extracted resonance frequency and the decay constant are displayed. a VCSEL device described in the literature [123] is considered. The simulated structure consists of 20 Bragg pairs in the bottom mirror and 15 Bragg pairs in the top mirror. Figure 5.7 shows both the device structure and the optical field of the fundamental mode. The metal contacts are isolated against the semiconductor mirrors at the mesa wall by a S13N4 layer. The contacts form an aperture for the optical field of d — 2.8 jum in diameter. In Figure 5.7 the simulation result for the fundamental mode shows the main features of interest in VCSEL design. The field is confined in the center of the device (1) where the intensity compared to the output beam (2) is higher by a factor of 30. From the top output beam a simple transformation can be applied to obtain the far-field pattern. The loss from the bottom mirror (3) can be seen as a wave propagating vertically from the center into the substrate. Radiating waves (4) are visible propagating at different angles from the center into the device. Furthermore, surface waves (5) propagate horizontally. These waves are guided by the layered medium. 94 CHAPTER 5. RESULTS Visualization of Simulation Light Output Results (Abs. Optical Field)x Tilted Mesa Substrate .. Figure 5.10: 3D Simulation of a tilted-pillar VCSEL. The top and bottom DBR mirrors consist of 16 and 23.5 periods, respectively The mesa was formed by dry etching the top DBR. A quadratic tilted pillar has been formed with a tilt angle of 20 %. Figure 5.8 displays the fundamental mode as well as the next higher order mode. The optical intensity is extracted from the time- domain data recorded in the points of the active region. The first eigenmode is ö>1/27r = 3.921 • 1014 + il.6 • 1011 Hz. As expected, this is near to the resonance frequency of the Bragg stacks, which are de¬ signed for a stop-band at the free-space wavelength A = 760nra. The next higher order eigenvalue is uj1/2tt = 4.625 • 1014 + i3.0 • 1011 Hz The shift to has been from idealized higher frequencies expected &ge¬ ometries [124]. The accuracy of the results can be estimated by the difference between the results for the Padé approximation for the different points in the active region. For the fundamental mode, the extracted poles he within ±0.03 % for the real part and ±2.5 % for the imaginary part of the eigenvalues. The simulation time was six hours on a Compaq Alpha Server ES40 with 667 MHz and the RAM requirement was 58 MB. A geometry variation has been performed in order to illustrate how the current approach can be applied to device optimization. With the 5.4. DISCUSSION 95 same mesa geometry, the aperture in the top contact has been varied. The fields of the simulations for the aperture diameter d — 2.8 p,m are compared to the results for d — 4.6 fiva in Figure 5.9. An FDTD simulation of a 3D VCSEL cavity has been presented recently [99]. The reduction from a 3D simulation domain to a 2D domain is desirable in order to lower the amount of computational resources needed. However, if the eigenmodes of complicated 3D VC¬ SELs have to be calculated, the presented method is the best choice. This is because the traveling-wave treatment of the electromagnetic field does not involve the inversion of a matrix. The problem scales linearly with the number of unknowns N. In contrast to this, the numerical treatment of (3.10) always involves the solution of a big al¬ gebraic eigenvalue problem. By using a volume discretization scheme, such as the finite-element method, the scaling of the computational cost is expected to be no better than N2, even if the sparsity of the matrix is exploited. Noting that if N is of the order of lOO'OOO for 2D and IOO'000'OOO for 3D, as it is the case for VCSEL structures, the solution of the algebraic eigenvalue problem will be difficult. As an example, a tilted-pillar VCSEL has been investigated in 3D, as shown in Figure 5.10. The tilted pillar was proposed to control the polarization behavior of the laser [125]. The mode degeneracy is removed due to the broken symmetry, and only one polarization is lasing. For the calculation, the numerical reflection coefficient of the FDTD Method has been modified [99]. As a consequence, a grid resolution of« A/12 can be used for the discretization of the structure. PML boundaries are used for the domain truncation. With the use of the phase correction scheme, the memory consumption for the 3D simulation is at about 2 GB. 5.4 Discussion In this chapter, a variety of optoelectronic devices have been simulated by the use of two distinct numerical methods. The FEM with edge- element expansion functions is used to solve eigenvalue problems in its most general form for both the waveguide and the cavity problems. With the use of FDTD, the simulation of the VCSEL eigenmode pat¬ tern as well as the calculation of spontaneous emission coupling have 96 CHAPTERS. RESULTS been demonstrated. Both methods have in common that they solve the electromagnetic problem in a rigorous, full-vectorial manner without restrictions for the geometrical structure of the device (except that a rotational sym¬ metry is required for VCSELs). Furthermore, both of the methods employ a volume discretization. For the purposes of device simula¬ tion, the volume discretization methods are superior to any boundary discretization method such as the boundary element method (BEM) or a multi-pole expansion method [126, 127]. This can be justified by the feature size of the optoelectronic devices, that is comparable to the optical wavelength A. The high number of A/4 layers in a VC¬ SEL, for example, make a boundary discretization impractical while a volume discretization is a natural choice. Further advantages become apparent if a temperature-dependent refractive index is considered. Any method which assumes piece wise constant material properties will be limited for this application. Also, good absorbing boundaries exist for both the FEM and the FDTD method. This is of vital im¬ portance since the mirror, diffraction and absorption losses are to be calculated. Despite the similarities discussed above, the FEM and the FDTD method are very different in the application to optoelectronic device design problems. Evaluation of the FEM The FEM is the method of choice for the coupling to an electro¬ thermal device simulator. It has been shown in Sec. 4.6 how the cavity and the waveguide solution fit into the equations for the com¬ prehensive simulation of semiconductor lasers. It has been shown that if some effort is spent on the solution of the resulting algebraic eigen¬ value problem, the performance of the FEM is very good. The huge problem sizes imposed by general VCSEL cavities (device diameters of ~ 100 wavelengths) can be solved within minutes on today's comput¬ ers. The performance depends very much on the advance knowledge about the structure, in particular, the resonance frequency has to be entered as a target value for the solution of the eigenvalue problem. This requirement is fulfilled for a coupled opto-electro-thermal simula¬ tion when the optics is re-iterated with the electronics. The accuracy 5.4. DISCUSSION 97 of the results is usually limited because the material parameters are not well known. Especially the optical absorption is not generally known, and a coupled opto-electronic scheme, combined with exten¬ sive measurement will bring much insight into this topic. Evaluation of the FDTD Method The FDTD method has been applied as an alternative tool to the FEM. As an example, the eigenpair of a microcavity resonator could be approximately derived by the FDTD method without any advance knowledge of the optical behavior, while the FEM can only be ap¬ plied with the advance knowledge of an eigenvalue. The flexibility of the FDTD method is its greatest advantage. The application range include the calculation of cavity resonances, coupling and reflection coefficients, absorption behavior such as the skin-effect, derivation of the optical mode density and many more. FDTD outperforms the other methods especially in the small memory requirement, since the numerical scheme never inverts the system matrix. As a consequence, full-vectorial 3D calculations as presented in Figure 5.10 are possible. Furthermore, FDTD provides a good accuracy control of the results. The primary error source for optoelectronic applications is the error introduced by the spatial grid that has to be imposed on the structure. A spatial resolution of at least A/40 is recommended if the phase er¬ ror has to be controlled due to layered media or distributed feedback (see Sec. 5.3.2). However, for the derivation of the frequency-domain results by the Padé approximation, the accuracy is not easy to con¬ trol. The automatic choice of the right number of poles and zeros is often difficult. In the case of the VCSEL device as presented the relevant eigenmodes are the one that survive longest in a time-domain calculation and can therefore be separated from the others. Chapter 6 Conclusion and Outlook 6.1 Major Results In this thesis, a concept has been worked out for a numerical solution of the optical fields and their coupling to an electro-thermal simulator. An optical mode solver has been implemented, and an interface to the existing device and circuit simulator DESSIS-ISE has been built. The range of devices cover edge-emitting lasers (EELs) as well as vertical-cavity surface emitting lasers (VCSELs). The main advan¬ tage of the presented approach is that all geometrical details can be included in the solution of the optical as well as the electronic model. With the resulting simulation tool, it has been shown that the concept of a physics-based device simulator is feasible for optoelectronic de¬ vices. Physical effects such as spatial hole burning or thermal lensing can be simulated with the fully-coupled models. Further findings include the traveling wave simulations of the op¬ tical fields. By the use of the finite-difference time-domain (FDTD) method, the optical behavior of the devices have been investigated in a complementary way. It has been shown that the FDTD method is an important and extremely versatile tool for the investigation of optoelectronic devices. A link has been established to the lumped-element laser mod¬ els. The parameters in the equivalent circuit models can be obtained 99 100 CHAPTER 6. CONCLUSION AND OUTLOOK by measurement. The presented device simulator provides a second source for the improvement of the empirical models. Parameters such as the confinement factor, the quantum efficiency or gain saturation can be investigated by a more rigorous, multi-dimensional simulation, and the results can be used in the equivalent circuit model. A method¬ ology has been presented for the integration of physics-based device simulation and measurements. Some more theoretical investigations had to be performed, and the approximations made in the eigenmode laser equations have been presented. Furthermore, the calculation of the optical mode in the context of the semiconductor device simulation had to be clarified. The differences between the waveguide-type eigenpairs as opposed to the cavity-type eigenpairs have been presented, and the applications have been brought in a relation to the mathematical foundations. 6.2 Further Development This work has not been carried out in isolation. Within the Op¬ toelectronics Modeling Group, the discussion of current and future development is essential for a long term planning. This is particu¬ larly important for the implementation of the models in DESSIS-ISE and for the cooperation with the project partners. In the following, remaining challenges in optoelectronic modeling are discussed with regard to the work presented in this thesis. Coupled opto-electro-thermal simulation of VCSELs A concept for the coupling of the optical modes and a semiconductor device simulator has been proposed in this thesis. It has been im¬ plemented for the simulation of edge-emitting lasers, and the scheme proved to be both versatile and robust. However, for VCSEL de¬ vices, only separate simulations of the optical problem and the electro¬ thermal problem has been done. The implementation of the coupling is challenging. A coupled opto-electro-thermal VCSEL simulator is expected to have a high impact on device development since optical mode control, current confinement and thermal management all are crucial design issues. 6.2. FURTHER DEVELOPMENT 101 Several scientific problems related to the electro-thermal solution are still open, e.g. to find an efficient but accurate way to solve the electro-thermal problem in the Bragg mirrors or to formulate the op¬ tical absorption comprehensively and in a way that is suitable for implementing it in a fully-coupled simulator. DFB and Multi-Section Laser Simulation In the multi-dimensional laser modeling, the advantages of the physics- based approach to device simulation is not yet fully exploited. The challenge for 3D self-consistent simulation of DFB lasers is in the ro¬ bust coupling between electronics and optics. The carrier distribution is influenced by spatial hole burning. The effect on the DFB spectrum is of practical importance and can be simulated with a physics-based multidimensional laser simulation. Furthermore, the presented approach is well suited for the simula¬ tion of multi-section lasers. With the use of multiple sections, tunable lasers can be realized, and ultimately, complete photonic integrated systems can be simulated. Of course, such devices have to be simu¬ lated in a fully-coupled manner, which is a challenging task for the future. Link between Device Simulation and System Simulation For the future, it is important to have a means to embed the de¬ vice simulation into a numerical treatment of the device surroundings. Both for integrated photonic circuits as well as for systems built from discrete devices, there is a need for a system simulation. As a con¬ sequence, an interface has to be provided not only for the electronic and thermal contacts, but also for the optical interconnection. It has to be possible to link several optical components, as for example in a complex telecommunication system. A survey has been performed on this topic, with the result of a methodology to extract electronic characteristics from laser simula¬ tions [128]. However, more work has to be done in this area. For the electronic and the thermal behavior, it is standard to embed the device simulation into an equivalent circuit, and the means for it is provided for example in the device and circuit simulator DESSIS-ISE. 102 CHAPTER 6. CONCLUSION AND OUTLOOK At the moment, similar possibilities are not available on the optical side. Some research has to be done to fill this gap. An interface may include the coupling between the device and an optical fiber, external optical feedback, or the crosstalk between highly integrated optical devices. As a result, it has to be possible to perform a hierarchical simulation that contains both the device physics as well as the system information. Calibration The implementation of the models is one thing. Finding the right ma¬ terial parameters for the simulation is a task by its own. Only with proper calibration of the models, the physics-based device simulator becomes a predictive device simulator. A methodology has been pro¬ posed in which it is shown that an advanced device development have to include both simulation and measurement. As suggested in Figure Figure 1.3, the gathering of material pa¬ rameters, the design of test series, and the extensive measurement are tasks typically performed inside of a company. In consequence, the material database is owned by the company. For the commercial success of the device simulator, this is essential and the ISE TCAD environment supports this concept. The project partners of this work have already started with improving their material data [129]. As a consequence of the above discussion, and as the Optoelec¬ tronics Modeling Group has grown over a critical size, it is planned to build a characterization laboratory, in which the research can be extended to cover material related topics and measurement. Refinement of the Electro-Thermal Models For the description of the optical fields inside semiconductor lasers, the solution of the Maxwell equation is the right choice, and a rigorous solution is possible and has been presented in this work. However, for the electronics, the situation is more complex. It is a challenge to choose an approximation to the fundamental equations. The solution of the Boltzmann transport equations, or (when quantized states are present in the device) even the many body Schrödinger equation is out of reach. Open questions remain on how to account for some 6.2. FURTHER DEVELOPMENT 103 effects observed in advanced optoelectronic devices. Examples are the spectral hole burning or quantum-wire and quantum-dot lasers. For these devices the assumption of quasi-equilibrium does not hold everywhere in the laser device, and an extension of the current models is necessary. Furthermore, the appearance of new materials for optoelectronic devices and the advances in process technology have to be accounted for on the simulation side. Only with a numerical treatment of bulk and quantum well carrier transport can these future requirements be met. Appendix A Maxwell's Equations Time-Domain Differential Representation V x E = -dtB (A VxH = dtD+j (A VB = 0 (A V D = g (A Integral Representation é Edl = -dt // B dS (A Jc JJs I H dl = dt [[ DdS+ ff j dS (A Jc JJs JJs B dS = 0 (A 's DdS= [[[ gdV (A 'S JJJv 105 106 APPENDIX A. MAXWELL'S EQUATIONS Frequency-Domain Representation V x E = iuB (A.9) VxH = -iujD+j (A. 10) V-B = 0 (A.ll) VD = q (A.12) Constitutive Relations D = e E (A.13) B = p H (A.14) In the case of inhomogeneous and dispersive media, s = e(r,uj) and p = ji(r,uj). In this case, the relations in the time-domain correspond to convolutions, as discussed in Appendix B. For anisotropic materials, s and p are 3x3 tensors. Bianisotropic media need an extension of the above formulae in order to relate D to E and H, as well as to relate B to H and E. Appendix B Semiconductor Dispersion Relations Introduction In the Maxwell equations, the whole influence of the material on the optical fields is contained in the permittivity e and the permeability p. For typical semiconductor materials, the permeability is constant, p = Pq. What seems elementary in the elegant notation of the Maxwell equations becomes sophisticated when it comes to write down the complete physical relationship. 1. It is clear that there is a spatial dependence, e(r). This makes an analytical solution of the Maxwell equations impossible except for very simple situations. 2. It has to be pointed out that the permittivity is complex, e G C. The imaginary part is responsible for material gain or loss. 3. The permittivity is frequency dependent s(uj). However, real and imaginary parts are not independent functions of uj. The additional relations are the Kramers-Kronig integrals summa¬ rized below. 107 108 APPENDIX B. DISPERSION RELATIONS Refractive Index at a Fixed Frequency For the coupled electro-opto-thermal simulation, the frequency of the optical field is defined by the optical cavity (lasing frequency uji). The complex permittivity e(r,uji,T,n,p), or equivalently the com¬ plex refractive index n(r,uji,T,n,p), are material parameters. The relationships are summarized in the following. e = e' ± ie" = (ri ± in")2 (B.l) e' = ri2 - ri'2 (B.2) e" = 2n'n" (B.3) ri = Re (vV ± ie"} = J^ (e' ± vV ± e") (B.4) ri' = Im L/e' ± ie"\ = J^ (-e! ± y/e' ± e"\ (B.5) The dependence on frequency, temperature and carrier density have to be indicated explicitly and prior to the device simulation. The dependence can be obtained from measurement and comprised in a look-up table for the relevant frequency interval and the expected temperature range. Typically, this data can be approximated by a polynomial. To account for material dispersion, the dependence on the angular frequency uj is linearized at the lasing frequency, ds(r,uj,T,n,p) (B.6) duj Kramers-Kronig Relations If a broad frequency range is considered, the real and imaginary part cannot be treated independently. The complex function e(uj) can be determined by the analysis of the underlying physics. For this aim, effects such as bandfilling, free-carrier absorption and band-gap shrink¬ age have to be modeled using analytical functions for the distribution functions for electrons and holes [3, 43]. 109 Starting from the time-domain Maxwell's equations, the polariza¬ tion P(t) can be introduced to account for the time-domain response of the material to some induced electric field, D(t) = e0E(t) + P(t) (B.7) P(t) = s0 I X(t ~ r)E(r)dr. (B.8) J — oc The function %(£) can be understood as the response of the material to a Dirac pulse. In order to retain causality, %(t) = 0 for t < 0. Applying the Fourier transform of the time-domain equations, the permittivity s(uj) can be related to %(£) by = (l ± \ e(uj) e0 J°° x(r)é"TdT, (B.9) with the complex angular frequency uj = uj' +iuj". Real and imaginary parts of s(uj) are related by the Kramers-Kronig integrals, e'(uj) = eo + -V ^^idû (B.10) 7T J0 UJZ - UJ1 s^)=.^vrm^^ (B.n) -K Jo UJ2 ~ UJ2 The change in refractive index for band-filling and bandgap shrinkage can be written as 2cfr * A „ ^ f°° Aa(n,p,E') An'(n,p, E) = —V J ^ ^ ;dE, (B.12) where the function a(n,p, E) can be calculated from the distribution functions fG and fv as well as the matrix element M, the effective masses me, rrih, and some fundamental constants. Appendix C Coupled Semiconductor Equations A semiconductor device can be modeled by the solution of a set of differen¬ tial equations. Typically, there is only a small number of primary variables (e.g. the carrier densities and the electric potential). The resulting equa¬ tions are often elaborate and some terms can be separated and often have a physical interpretation on its own (e.g. the current densities or the recombi¬ nation coefficients). When solving the system of equations, these functional relations are substituted back into the differential equations, and the system is written in divergence form V • F(r) - s(r) = 0. (C.l) Once the primary variables have been identified, it is in the advanced func¬ tional dependencies (e.g. the recombination and mobility models) where the physical effects have to be implemented. Basic Semiconductor Equations In order to simulate semiconductor material with different doping levels Np and N^ the simplest set of primary variables consists of the electron and hole density and the electric potential. In order to have only first order differential equations, the electric displacement D has been introduced. Equation (C.5) can be substituted in (C.2), so that in total, only three substantial differential equations result. Note that the recombination terms 111 112 APPENDIX C. COUPLED SEMICONDUCTOR EQUATIONS have units [1/to ] and can be written as the sum of several recombination processes. Carrier generation is equivalent to a negative recombination. V.D = -g(p-n + AT+-iV-) (C.2) V jn =q(R + dtn) (C.3) -V-jp = q(R + dtp) (C.4) D = eV n = NcF^ (C.6) { kBT ) + AT fq$p Ev\ P = (C.7) NvFl/2{ kßT j f°° sjxdx (C.8) Fl/2(r,)=-yJo 1 + eX-v jn = -qßnnV Jp = -q/J>PpV(f) - qDpVp (CIO) Thermodynamic Equations For the thermodynamic simulation, the lattice temperature T is introduced as a new primary variable, and the additional differential equation is the heat equation. The current relations (C.9) and (C.IO) are replaced, and new relations for the thermoelectric power for the electrons and holes Pn and Pp are introduced: V • S = -ctotdtT - V • (jn (PnT + $„) +jp (PPT + $p)) (C.ll) S = -Kth^T (C.12) (C.13) (C.14) jn = —q (unnV(j) — Dn'Vn + ßnnPnVT) (C.15) jp = -Q {p>PpV Hydrodynamic Equations In hydrodynamic simulation mode, separate variables are introduced for the lattice temperature, and the carrier temperatures. The thermodynamic equations are replaced by V • SL = -cLdtTL + RE, nTn — nTL — 3&B '"" — pTL (nt^m , __rr< \ , , pTp + % (R(nTn+PTp) + + C.17) V-Sn= jnVEc - ^ (dt (nTn) - RnTn - C.18) V • Sp =jpVEv - ^ (dt (pTp) - RpTp - C.19) C.20) nTnVTn C.21) Ï 2 5r / kB . c rr, , kBßn rr r7rr\ = [ SP —7T —Tpjp H pTpVTp C.22) 2 V jn = ßn (nVEc ± kßTnVn — 3/2nkBTnV\nme) C.23) jP = ßp (pVEy + kßTpVp — Z/2pkßTpVln.mh) C.24) 2D Fabry-Perot Laser Equations In the laser simulation mode, the photon rate equation is the key exten¬ sion in the semiconductor equations. The eigenvalue equation of the optical problem have to be solved to get the optical modes. For multi-mode sim¬ ulations, an additional photon rate equation is introduced for each mode v. The variable pe (r, E') is the density of states of the charge carriers. In bulk semiconductors, pe (r, E') is proportional to the square root of the en¬ ergy difference to the band edge. The spontaneous emission coupling factor ß is discussed in Sec. 2.4.2. For the calculation of local material gain rst and the local spontaneous emission rsp, the broadening function L(E, E') is employed. Possible choices for L(E, E') include the well-known Lorentzian broadening [42]. dtS„ = vg (G„ - U) S„ + vgßvKzRtv (C.25) M 7, <- (v2 + ^-er(r) - 7') M*0 = 0 (C.26) 114 APPENDIX C. COUPLED SEMICONDUCTOR EQUATIONS UJl = max ff rst(r,hu)\§„(r)\2d2r C.27) Gv = JJrst(r,E)\E=huji\^(r)\2d2r C.28) JlnWR2+ImM C.29) (VRi + VW){1 - VR1R2) Kz C.30) VR^M\n(l/(R1R2)) Rtp ffrsp(r,E)\E=nuji\^(r)\2d2r C.31) rS\r,E) J2 f C0|M^|Vi(r,£')(/f(r,£')±/,V(r,£')-l)L(£,£')d E' C.32) rSP(r,E: = J2[ Co|MM|V \r,£')(/f {r,E>)ff{r,E>))L(E,E')dE' C.33) -c + = U eTWri^+^n / + ^E)^-1 C.34) tV + = ^^Y1 r U + eTWri^+^. C.35) Appendix D Notation Conventions The same character is used for variables with different dependencies, e.g. a(t) and a(uj). To enhance the readability, the notation a(t) is sometimes used in large formulas. Harmonic time dependence is chosen as e~%ujt, and the propagating wave expansion is chosen as e-lljjt+liz. The coordinate systems for edge-emitting and surface-emitting structures is displayed in Figure 1.1 and Figure 1.2, respectively. Po¬ lar coordinates are introduced for the analysis of circularly symmetric structures in Figure 3.1. Variables (.)T Transpose of a vector * a Complex conjugate. In formulas also denoted by 'cc' i h a a Real and of a — a ia" , imaginary part complex variable, -\- \a\ Absolute value of a complex variable or a vector La Angle of a complex variable = a Vector a (ai, a2, ... an) [A] Matrix [A\l3 — al3 Partial time — dt derivative, dt -qi dn Spatial derivative in normal direction 115 116 APPENDIX D. NOTATION dV Closed surface dS Bounding contour V Nabla operator. In Cartesian coords., V = (dx, dy, dz)T Vt Transverse nabla operator. In Cartesian coords., Vt = (dx, dy) (.)t Time average A Vector potential B Magnetic flux density c Speed of light in free space C Capacitance C Complex numbers D Electric displacement flux density D Diffusion constant e Unit vector E Energy Eg Energy gap Ec, Ey Conduction and valence band edge E Electric field Ex Component of electric field in ar-direction (likewise for y and z) E Longitudinal optical field pattern (scalar, complex) Er,El Envelope of right and left traveling waves (scalar, complex) T{.} Fourier transform fn, fp Distribution function of electrons and holes / Frequency F Cavity finesse G Total optical gain [1/m] g Incremental optical gain [1/m,] gd Small-signal conductance Q Green's function H Magnetic field h Reduced Planck constant, h = 1.05459 • 10"34 Js H Hamiltonian / Current — i Imaginary unit, i — 1 Im (.) Imaginary part of a complex number j Current density k Wave vector k Wave number, k — \k\ • kß Boltzmann constant, ks — 1.3807 10~23J/K L Optical loss [1/m] £ Length me, m,h Electron and hole effective mass 1 mo Free electron mass, mo = 9.1095 • 10~31 kg n Electron density n Complex refractive index, n — n' + in" Np, N^ Activated donor and acceptor concentrations P Thermoelectric power V Principal value p Hole density Q Resonator quality factor • q Elementary charge, q — 1.60219 10~19 C r (Amplitude) reflectivity rd Small-signal resistance R Ohmic resistance rst Local material gain [1/m] rsp Local spontaneous emission [1/m] r Location in space r = (x, y, z) R Recombination Rsp Total spontaneous emission rate R„p Spontaneous emission rate into optical mode v Rcap Capture rate Re (.) Real part of a complex number R Real numbers S Photon number (scalar, real) s Scalar field S Surface of a geometrical object S Power flux T Temperature t Time V Normalized frequency V Voltage. Sometimes also: Volume of a geometrical object v (Phase) velocity — vg Group velocity, vg(X) — v(X) X vdA Z Complex impedance a Decay constant ßv Spontaneous emission coupling factor, ßv — R^pjRsp 7 Complex propagation constant, 7 = 7' + ij" S Dirac distribution function A Laplacian operator. In Cartesian coords., A = d2 + dy + d2 At Transverse Laplacian. In Cartesian coords., At — d2 + d2 e Electric permittivity in electronics er Relative permittivity, e — er£o e Complex electric permittivity in optics, e — e' + is" 118 APPENDIX D. NOTATION e, Relative permittivity, e, — ereo, £o — 8.8542 10~12 C/(Vm) £ Phase of the 3D optical field pattern (scalar, real) n Efficienty 0 Amplitude of the 3D optical field pattern (vector, real) $ Relative refractive index variation (=weak-guidance parameter) Kth Thermal conductivity At Absorption coefficient, k — ri X Wavelength • p, Magnetic permeability, for optics: fj, — fio — 47r 10~ H/m v Mode index £ Eigenvalue in the algebraic eigenproblem popt Optical mode density pel Density of states for the electrons and holes p Location in transverse plane p — (x,y) a Electric conductivity t Characteristic time constant Tph Photon life time $ Transverse optical field pattern (vector, complex) $ Fermi potential $e, &h Quasi-Fermi potentials for electrons and holes (j) Electric potential X Electric susceptibility \I/ 3D optical field pattern (vector, complex) oo Angular frequency 119 Abbreviations ABC Absorbing Boundary Condition AR Anti Reflective (Coating) BEM Boundary-Element Method BOR Body of Revolution BPM Beam-Propagation Method CGSEL Circular-Grating Surface-Emitting Laser CPU Central Processing Unit CW Continuous-Wave DFB Distributed-Feedback Laser DBR Distributed Bragg Reflector EEL Edge-Emitting Laser ETHZ Eidgenössische Technische Hochschule Zürich FDTD Finite-Difference Time-Domain Method FEM Finite-Element Method FFT Fast Fourier Transform FP Fabry-Perot Laser HR Highly Reflective (Coating) IIS Institut für Integrierte Systeme ISE Integrates Systems Engineering AG LASER Light Amplification by Stimulated Emission of Radiation LED Light-Emitting Diode PEC Perfect Electric Conductor PMC Perfect Magnetic Conductor PML Perfectly Matched Layer R&D Research and Development RCLED Resonant-Cavity Light-Emitting Diode SLLED Super-Luminescent Light-Emitting Diode SOA Semiconductor Optical Amplifier TE Transverse Electric TM Transverse Magnetic TMM Transfer-Matrix Method TCAD Technology Computer Aided Design UCSB University of California, Santa Barbara VCSEL Vertical-Cavity Surface-Emitting Laser WDM Wavelength-Division Multiplexing Bibliography [1] L. 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From 1995 to 1997 he worked as a part- time employee at the IBM Research Laboratory in Rüschlikon, Zürich, where he was involved in the fabrication of 980 nm pump lasers. In 1997 he joined the Integrated System Laboratory as a research and teaching assistant. In the first year, he worked on the development of a finite-difference time-domain solver for electromagnetic wave prob¬ lems. Since 1998 he works in the Optoelectronics Modeling Group on numerical simulation of semiconductor lasers and optoelectronic devices. In 2002 he was an academic research guest at the University of California, Santa Barbara, a work supported by a scholarship of ETH. His research interests are in the field of numerical modeling of optoelectronic devices. 131\2d2r. (2.56)(x,y) (Ef(z,t)e-iut+i^ E^ty-***-**) , (2.70) with a transverse field pattern $ and the forward and backward traveling wave Ef(z,t) and Eb(z,t), respectively. Substituted into Maxwell's equations, the following laser rate equations can be derived: