| PREFACE
The book is divided into nine chapters. Except for the first two introductory chapters, each chapter is independent and restricted to a particular subject to be studied. To the best of the author s knowledge, the most appropriate theories have been chosen to model the specific topic of VCSELs. In Chapters 3 and 4, theoretical models have been developed to analyze the modal profile and polarization, respectively, of VCSELs. The most popular structure of VCSELs is a cylindrical symmetric cavity, which is assumed in the derivation of the models. In addition, this configuration of VCSELs allows investigation of the modal profile and polarization separately such that the complexity of theoretical models can be reduced. In Chapter 3, different methods of solving the wave equation for the modal profile of VCSELs are discussed in detail. The reader can choose the most appropriate model with the required speed and accuracy to analyze the problems. In Chapter 4, two- and four-level models are described to study the polarization properties of the fundamental transverse mode. These simplified models can evaluate the dominant factors that control the polarization properties of VCSELs. It must be noted that the investigation of VCSELs using cold cavity approximations is not realistic. This is so because most of the measurable data, such as threshold current, lasing wavelength, slope efficiency, and output power, all depend on the operating temperature of lasers. Furthermore, the optical behavior of VCSELs is affected by thermal lensing (i.e., self-focusing of transverse modes into the core region of the active layer). Therefore, the thermal properties of VCSELs are investigated in Chapter 5. The method of effective temperature using a simple rate equation model is presented. Effective thermal conductivity and heat generation rate are also derived. The objective in defining effective temperature is to simplify the study by using a rate equation model so that the computational efficiency can be improved. However, this approach will not provide detailed information on heat distribution. Detailed heat distribution inside the laser cavity is studied by solving the heat equation numerically. In this case, the influence of thermal lensing on the optical field profile can be evaluated. Spatial hole burning of carrier concentration also has significant influence on the modal profile of VCSELs. Therefore, Chapter 6 describes the use of a simple rate equation to evaluate the distribution of carrier concentration inside the active region. In this case, self-consistent calculation of optical gain and carrier concentration (i.e., self-consistent calculation of the Poisson and Schrödinger equations) is ignored to simplify the calculation. Different methods for approximating the nonuniform distribution of carrier concentration are also discussed. On the other hand, nonuniform distributions of electric potential and current are required as the input parameters to calculate the heat distribution inside the laser cavity. They have to be solved numerically using the Poisson and continuity equations simultaneously with appropriate boundary conditions. The electric potential across the active layer and the corresponding carrier concentration can be linked together by a simple diode equation. This is so because the simplified relation between optical gain and carrier concentration has been assumed. The self-consistent calculation of optical field, heat, and electrical characteristics of VCSELs is also described in Chapter 6. The dynamic response of VCSELs is analyzed in Chapter 7. Preliminary investigation of the dynamic response of VCSELs using a simple rate equation model is described. Hence, the time variation of carrier concentration and photon density inside the active layer can be calculated. Furthermore, detailed analysis of optical fields can be considered using the beam propagation method such that the influence of optical confinement on the dynamic response of VCSELs can be evaluated. However, detailed investigation of the transient response of heat and electrical properties is avoided in the self-consistent calculation. This is because the time variation of heat and voltage, which are related to heat and the Poisson equations, is much slower than that of photon density and carrier concentration. This assumption significantly reduces the computation time of the model without sacrificing the accuracy of the calculation. The influence of various transportation mechanisms inside the quantum well (QW) active region on the dynamic response of VCSELs is also discussed in this chapter. The methods used to evaluate the spontaneous emission and linewidth of VCSELs are described in Chapter 8. Simple models have been developed to study these parameters quantitatively through the investigation of the spontaneous emission factor and linewidth enhancement factor. On the other hand, the magnitude of the spontaneous emission factor and linewidth enhancement factor is evaluated using rate equation model by empirically fitting the measurable data. Hence, design criteria to optimize the spontaneous emission of VCSELs are obtained. Other nonlinear features of VCSELs such as self-sustained pulsation, bistability, dual-wavelength operation, and wavelength tunability are studied in Chapter 9 using rate equation models. The advantage of using simple rate equation models is that the parameters that describe the nonlinear behavior of VCSELs can be easily extracted through some measurable data such as injection current and lasing power. In conclusion, this book presents the most effective way to implement laser models of VCSELs, which the reader can easily understand. However, the readers are assumed to have the usual undergraduate background knowledge of electromagnetic theory and solid-state physics as well as basic computational skills. Materials of this research monograph concentrate on the evaluation of modeling techniques to analyze VCSELs under various operating conditions. As each chapter of this book is mostly independent of the other chapters, readers can selectively study any chapter for their own interest. Although this book is of most interest to the design engineer of VCSELs, it also provides valuable information to CAD tool designers in other fields of semiconductor lasers. Siu Fung Yu Singapore |
Design and fabrication of vertical cavity surface emitting lasers (VCSELs) requires an iterative process, which is extremely expensive and time-consuming. The use of computer-aided design (CAD) tools can help shorten the design cycle and speed up the development process. Laser models, which are found in the literature, can be used to implement CAD tools for the analysis and design of VCSELs. However, some comprehensive models, which perform sophisticated functions, are difficult to implement and show low computational efficiency. Other simplified models exhibit high computing speed but deliver inadequate descriptions of the observed effects. As a result, inconsistent conclusions may be obtained because different assumptions are applied. This book attempts to provide a guideline for the derivation of models based on appropriate assumptions for a particular problem so that the phenomena observed by the experiment can be easily explained. In fact, the objective throughout this book is to search for the simplest and most direct treatment for modeling VCSELs. The author believes that the laser models covered in this book can help the readers customize their CAD tools to fit into their applications. In addition, the readers should have no difficulty in implementing their own laser models.
TABLE OF CONTENTS Chapter 4A.2 - Band Structure Model
4A.2 Band Structure Model In order to calculate the electron and hole wavefunctions in QWs, the multiband effective mass theory is used [22]. For most III-V semiconductor materials sucl as GaAs-based materials, it is a good approximation that the conduction and valence bands are decoupled. A parabolic band model and a Luttinger-Kohr Hamiltonian with strain components are used to calculate the conduction and valence bands, respectively [22]. The electron states near the conduction subband edge are assumed to be almost purely s-like and nondegenerate (excluding spin), while the hole states near the valence subband edge are almost purely p-like and fourfold degenerate (including spin). The envelope function scheme is adopted to describe the slowly varying part of the wavefunction. In the following paragraphs, the slowly varying part of the wavefunction for the electron and hole are calculated. The influence of QW confinement potential on the energies and envelope functions of the electron subband edge at the zone center of the Brillouin zone can be calculated separately using the one-dimensional Schrödinger-type equation:  where The valence band structure in QWs is more complicated as there is a fourfold degeneracy (including spin degeneracy) at the top of the valence band. The periodic part of the Bloch function (not including the spin degeneracy) at the top of the valence band has the symmetry of a p-type wavefunction that is threefold degenerate. Combining with the spin, there are six valence bands (the heavy hole band, the light hole band, and the spin-orbit splitoff band) just below the conduction band. The spin-orbit splitoff band is split from the heavy hole and light hole bands by the spin-orbit interaction. In addition, the heavy and light hole valence bands are split as a result of the quantum confinement effect. If the energy separation of the spin-orbit splitoff band is far away from the heavy and light hole bands, the corresponding envelope function can be obtained by  where Ψvl is the envelope function of the lth subband for holes,  and the parameters P,Q,R and S are given by   where γ1, γ2, γ3 are the Luttinger-Kohn parameters, which are dependent on the position z and mo is the relative mass of electron. The Luttinger–Kohn Hamiltonian is actually a set of coupled linear differential equations for the envelope functions, which can also be solved using the finite difference approximation. The periodic part of the Bloch state (basis function) for electrons is given by  where ↑ and ↓ denote the up and down electron spinors and s  When the subband envelope functions are obtained, the optical matrix elements can be calculated by the following expression  where p and q denote the Bloch functions of electron or holes and υp  where |
, Ψcl is the envelope function of the lth subband for electrons, m* is the effective mass in the z direction, Ecl is the subband edge energy, and Uc is the QW confinement potential of electrons. This equation can be solved numerically using a finite difference method with the corresponding confinement profile for the approximated parabolic band.
is the Luttinger-Kohn Hamiltonian with strain introduced, and Uv is the confinement potential for hole, which are given by
is the s-like conduction band Bloch states. For holes, the Bloch states are represented by the linear combinations of the products of the spinor and the ρ-like valence band Bloch states px
Ψcp Ψvs,q 
, 
) is the overlap integral of the envelope functions and the expression of Ψvs,q is obtained from the one-dimensional Schrödinger type equation (4A.5).
