Numerical analysis of supermodes, modal gain, and differential gain in heterogeneously integrated InP/Si lasers

Transcription

1 Numerical analysis of supermodes, modal gain, and differential gain in heterogeneously integrated InP/Si lasers Chen-Kuo Wu Promoter: Prof. dr. Geert Morthier Supervisor: Amin Abbasi Master's dissertation submitted in order to obtain the academic degree of European Master of Science in Photonics Department of Information Technology Chair: Prof. dr. ir. Daniël De Zutter Faculty of Engineering and Architecture Academic year !

2 Preface IamreallyluckythatIhavethisopportunitytojoinEMSPwhichisprovidedbyPhotonics Group in Universiteit Gent and Graduated Institute of Photonics and Optoelectronics in National Taiwan University. Also, I want to thank Prof. Yuh-Renn and Hsiao-Wen in NTU for encouraging me to apply for this program. Without your encouragement, I would not dare to live, study, and do the research alone in a foreign country. Studying in Gent was very ambiguous to me before arriving here. I was afraid of everything at that time. Talking to foreigners, following the course, looking for accommodation or applying for the residence card made me feel stressful. But, luckily again, my promoter, Prof. Geert Morthier, and my supervisor, Amin Abassi, are really kind to help me and guide me. Whenever I had questions, you spent time with me discussing how to solve the problems, and also paying the patience to teach me how to write the master thesis. I am not good at presenting my emotion, but I want to sincerely say Thank you! to Prof. Geert, and Amin. On the other hand, my Belgian friends, Ar Anit, Bert, Simon, and Wendy, you guys are really friendly to me and help me a lot. Even though my English is not that well, you are still willing to listen to me and assist me to overcome lots of annoying problems in my daily life. These are really meaningful to a person who never left his home so long. Also, I would like to thank my Taiwanese EMSP partners, Fang-Ching, Kang Hsien, Ting-Hao, Shih-Chang, Yen-Hsiang, and Yi-Weng, and all my friends in Taiwan for the social support. Lastly, I would like to leave this space for my parents. You work hard for providing me a good environment, letting me explore the world without hesitation, and guiding me to face the different problems properly. You always respect and support my decisions. Mom, Dad, I love you so much! Chen-Kuo Wu June 4 th 2016 i

3 Permissions The author gives his permission to make this work available for consultation and to copy part of the work for personal use. Any other use is bound to the restriction of copyright legislation, in particular regarding the obligation to specify the source when using results of this work. Chen-Kuo Wu, June 2016 ii

4 Toelating De auteur geeft de toelating dit afstudeerwerk voor consultatie beschikbaar te stellen en delen van het afstudeerwerk te copieren voor persoonlijk gebruik. Elk ander gebruik valt onder de beperkingen van het auteursrecht, in het bijzonder met betrekking tot de verplichting de bron uitdrukkelijk te vermelden bij het aanhalen van resultaten uit dit afstudeerwerk. Chen-Kuo Wu, June 2016 iii

5 Numerical analysis of supermodes, modal gain, and differential gain in heterogeneously integrated InP/Si lasers by Chen-Kuo Wu Master thesis dissertation for achieving the academic degree of European Master of Science in Photonics Academic year Promotor: Prof. dr. ir. Geert Morthier Supervisor: Ir. Amin Abbasi Department of Information Technology (INTEC) Prof. Dr. Ir. D. De Zutter Faculty of Engineering and Architecture Ghent University Summary Our goal is to achieve a high power and high speed 1550 nm distributed feedback (DFB) integrated InP/Si laser. We numerically analyzed the device performance, and in particular the modal gain and the mirror losses. We used AlGaInAs instead of the traditionally used InGaAsP for the material of the MQW active layers. The higher conduction band offset ratio of AlGaInAs (1.562 times larger than InGaAsP) guarantee higher confinement for hot electrons at high temperature. With the higher optical confinement in the active region, the injected carrier density can be lower for the same optical gain, making the differential gain and the operation speed also higher. Besides, in order to integrate the lasers on the Si platform, the confinement factor in Si should not be too low. The calculated results show that increasing the QW number can enhance the optical confinement in the active region. Furthermore, a thinner n-cladding layer, a thicker separate confinement heterostructure (SCH) layer, and a thicker bonding layer can increase the optical confinement factor in the active region except for QW number less than 6. Besides, we found that the BCB thickness should be less than 60 nm for having a good optical confinement factor in the Si waveguide. On the other hand, the evaluated coupling coefficients, which are related to the reflectivity of the DFB grating, show that a higher QW number, thicker n-cladding layer, thicker BCB layer, and thicker SCH layer in the devices result in lower coupling coefficients. The result implies that the cavity length should be longer in these devices to obtain the same reflectivity. With the help of the optical field calculations, we can further evaluate the optical gain and the differential gain of our devices. The simulations show that more QWs or higher optical confinement factors in the active region will result in a higher differential gain. The total volume of the QW layers and the differential gain are also considered in the calculation of the relaxation oscillation resonance frequency. The result shows that the 12 QW case might potentially give the highest speed in the real devices. Keywords: DFB high speed laser, hybrid III-V/Si, AlGaInAs, modal gain, relaxation oscillation resonance frequency iv

6 1 Numerical analysis of supermodes, modal gain, and differential gain in heterogeneously integrated InP/Si lasers Chen-Kuo Wu Supervisor(s): Geert Morthier, Amin Abbasi Abstract In this article, we use numerical models to explore the possible designs of a high speed and high power 1550 nm distributed feed back (DFB) integrated InP/Si lasers. AlGaInAs with high conduction band offset ratio is chosen as the material of the multiple quantum well (MQW) active layers. We aim for structures with good confinement factors in the InP active layer and the Si region. After considering the volume and the optical confinement factor in the quantum well (QW) layers, the calculated relaxation oscillation resonance frequency shows that the laser with 12 QWs might give the fastest speed. I. INTRODUCTION Optical fiber communications is regarded as necessary in the development of advanced cyber technology [1]. An optical communication system is commonly comprised of III/V semiconductor laser based signal sources and optical fibers, whereby the 1550 nm wavelength is the most popular wavelength due to its lowest attenuation in glass fibers. Traditionally, InGaAsP, which has the notoriously weak confinement of hot electrons at high temperatures, is used to be the material of the MQW active layers. However, AlGaInAs with a higher conduction band offset is proposed to replace the InGaAsP. Due to the compatibility with CMOS processing and the high index contrast, silicon photonics is receiving ever more attention and more and more photonic components are demonstrated in this material platform. Distributed feedback lasers have been demonstrated by heterogeneous integration of III-V membranes on silicon-on-insulator waveguide structures. For a good laser operation, sufficient optical confinement of the laser mode with both the active layer and the Si grating is required. Hence, we calculated the optical field and the relaxation oscillation resonance frequency for different thicknesses of the different epilayers, and see what compositions is most suitable for a high speed 1550 nm DFB laser. Fig. 1: Effective band diagram and the band offset of Al- GaInAs and InGaAsP based devices with the same bandgap of QW and QB. The emission wavelength is around 1550 nm. gain, and the relaxation oscillation resonance frequency. Those equations and the numerical methods can be found in the references. The modulation bandwidth is determined by the relaxation oscillation frequency fr, which can be expressed as: f r = 1 2π v g dg/dn en w d w W act L (I I th), (1) where v g is the group velocity, N w, d w, W, and L are the number of QWs, the thickness of a single QW, the width of the active region, and the length of the cavity. dg/dn is the differential gain, where the optical confinement in the total QW layers is considered. I and I th are the current and threshold current. II. SIMULATION METHOD We have integrated different models to evaluate the performance of the complex hybrid lasers. At first, the Schrödinger equation with k p method, the Poisson and drift-diffusion equations are used to analyze the electrical problems [2]. On the other hand, the distributions of optical fields in the III-V/Si hybrid waveguide are estimated [3]. In the end, the quantized energy levels calculated by Schrödinger equation and the optical confinement factors in the QW layers are considered into the calculations of the optical gain [4], the differential III. ENERGY BAND ANALYSIS Fig. 1 shows the energy band diagram of a single QW with AlGaInAs and InGaAsP by considering the strain effect and quantized levels. Both the bandgap of the QW and QB in are the same for both materials. The emission wavelength is around 1550 nm wavelength. The conduction band offset ratios of AlGaInAs and InGaAsP are and There will be a better connement of hot electrons at high temperature, giving less leakage current, in the AlGaInAs case.

7 2 Fig. 2: (a) The selected structures with 75 nm SCH, 10 nm BCB, 100 nm n-cladding layer, and different QW numbers. (b) The confinement factors of different QW cases. Fig. 4: The extracted modal gain values of the different QW number cases at 1550 nm wavelength. We assume that the threshold gain is 40 cm 1. Fig. 3: (a) The confinement factors with different SCH thickness, BCB thickness, and 100 nm n-cladding layer. (b) The confinement factors with 10 nm BCB and 75 nm SCH with different thickness of n-cladding layer. dg/dn Fig. 5: The extracted N wd ww actl values of the different QW number cases at 1550 nm wavelength. We assume that the threshold gain is 40 cm 1. A. QW number IV. OPTICAL FIELD DISTRIBUTION Figure 2 (a) shows the simulated structures with 75 nm separate confinement heterostructure (SCH), 10 nm Benzocyclobutene polymer (BCB), 100 nm n-cladding layer, and different QW numbers. Figure 2 (b) shows that increasing the volume of the active regions elevates the confinement factor Γ act. and decrease the confinement factor in Si Γ Si. In addition, the optical confinement factors in single QW of each case are %, %, %, where the 12 QW case shows the highest confinement factors per QW. B. Thickness of the SCH, BCB and n-cladding Layers In addition, the thickness of SCH, BCB, and n-cladding layer will also influence the optical field distribution in the hybrid waveguide. Figure 3 (a) shows that increasing the thickness of the SCH layers and decreasing the thickness of the BCB layer can increase the optical confinement factor in the active region and reduce the value in the Si region. But, both the confinement factors in the active region and Si decrease with increasing thickness of the n-cladding layer in Fig. 3 (b). The decreasing confinement factors in the Si region are attributed to the increasing volume of the III/V waveguide. However, the enlarged part of the n-cladding layer shares some of the optical fields with the active region, and results in a smaller confinement factor in the active region. Our further calculations (not shown here) reveal that the optimization of the 6 QW case by changing the different epilayers is limited due to an insufficient thickness of the active region. Nevertheless, the thickness of the BCB layer should be kept below 60 nm. Otherwise, the optical confinement factor of the Si region will be too low. V. RELAXATION OSCILLATION RESONANCE FREQUENCY We further extract the gain value at 1550 nm wavelength in Fig. 4, where the structures are the same as in Fig. 2(a). The result clearly shows that a higher QW number gives a higher modal gain value for a given carrier density due to its higher optical confinement factor in the QW layers. Also, a higher QW number case shows a lower threshold carrier density for a fixed total threshold gain (e.g., g th = 40 cm 1 ). Furthermore, we consider the differential gain, which is calculated by the data of the gain curves with different injected carrier densities, dg/dn N wd ww actl and the volume into the calculation of values from Eq. 1, since the speed of lasers is not only related to the differential gain but also the volume of the total QW layers. The cavity length L is µm. The result in Fig. 5 shows that the laser with more QWs still performs better. Even though the increasing number of QWs will increase the volume which slows down the operation speed, the high QW number case (12 QWs) still gives the highest modulation bandwidth due to its very high differential modal gain. VI. CONCLUSION We have reported on numerical calculations of the 1550 nm distributed feedback (DFB) integrated InP/Si lasers. AlGaInAs

8 3 is suggested to be the material of choice for the MQW active layers. A higher optical confinement factor in the active region will result in a higher speed. REFERENCES [1] C. Sun et al. Nature 528(7583), pp , [2] Simulase manual, [3] P. Design, Fimmwave, [4] V. Lysak

9 Contents Preface i Permissions ii Toelating iii Summary iv Contents v List of Figures vi List of Tables x Abbreviations Introduction Motivation Semiconductors Fundamental Physics of Semiconductors Energy Band in a Typical QW Device III-V/Si Hybrid Waveguide Semiconductor Laser Optical Gain Principle of Semiconductor Lasers DFB Lasers Thesis Overview Methodology Energy Band and Quantum Level Optical Mode Calculation Optical Gain Energy Band Analysis Energy Band Diagram and Strain Distribution Free Carrier Distribution Study of Optical Field Distribution Mode Selection in the Passive Waveguide Different Structures of III-V/Si Hybrid waveguide Optical Confinement Factor with Different QW numbers QWs QWs QWs Comparison of Confinement Factors with Different QW Numbers Coupling coefficient QWs QWs QWs Comparison of Coupling Coefficient with Different QW Numbers Optical Gain and Differential Gain Calculation Conclusion Energy Band Analysis Study of Optical Field Distribution Optical Gain and Differential Gain Calculation Bibliography v

10 List of Figures 1.1 Dispersion vs. wavelength in optical fiber [1] Attenuation vs. wavelength in optical fiber [2] A typical Fermi-Dirac distribution in a semiconductor system [3] (a) The density of states in the bulk semiconductor. (b) The density of states in the QW semiconductor. [4] (a) The Fermi-Dirac distribution in a n-type semiconductor system. (b) The Fermi-Dirac distribution in a p-type semiconductor system. [3] The scheme of a p-n QW device The illustration of the quantized energy states in the QW (a) The illustration of carrier behavior in a optoelectronic device. (b) The scheme of Auger recombination The concept of the light travels in a heteromaterial system [5]. (n f >n s ) The scheme of the TE and TM filed propagate in a planar waveguide [5] Different orders of TE mode in a planar waveguide [5] A typical III-V/Si hybrid waveguide [6] The calculation of the confinement factor for a even fundamental mode of the III/V active region and the Si from the group from California Institute of Technology [7] The Gain spectrum of unstrained 8 nm GaAs/Al 0.15 Ga 0.85 As QW/QB with different injection carrier density (10 18 cm 3 )[8] The band edge of compressive, unstrained, and tensile 8 nm GaAs/Al 0.15 Ga 0.85 As [8] The band edge of compressive, unstrained, and tensile 8 nm GaAs/Al 0.15 Ga 0.85 As. [8] The scheme of an optical cavity. R 1 and R 2 are the reflective index of the mirrors. Γg material and α are the optical gain and loss of the internal cavity, and the unit of both parameters is cm 1.Listhelengthofthecavity The scheme of a grating. n A and n B are different refrative index of the material. Λ is the period of the grating [5] Four typical optcal cavities applied in semiconductor laser. (a) Fabry-Perot Laser. (b) DBR laser. (c) DFB laser. (d) VCSEL. [9, 10] The scheme of a diffraction grating. [11] The concept of the finite difference method [12] The Gain spectra of 6 nm single Al Ga In 0.7 As/Al 0.23 Ga In As QW/QB with cm 3 injected electron and hole density under different relaxation times The simulated single QW and two QBs structures with different material systems Band diagram and quantum level of AlGaInAs and InGaAsP based devices Effective band diagram and the band offset of AlGaInAs and InGaAsP based devices The strain distribution along the z-direction of AlGaInAs and InGaAsP based devices vi

11 3.5 The free carrier distribution along the z-direction of AlGaInAs and In- GaAsP based devices The energy band diagram along the z-direction of the AlGaInAs device with different barrier compositions Al 0.23 Ga In As and Al 0.25 Ga In As The free carrier distribution along the z-direction of the AlGaInAs device with different barrier compositon Al 0.23 Ga In As and Al 0.25 Ga In As The simulated cross section in the DFB laser The structure and the refractive index in the simulated III-V/Si hybrid waveguide. W is the thickness of the n-cladding layer. X is the thickness of SCH layer. Y is the thickness of the active region (MQWs). Z is the thickness of the BCB layer (a) The example structure. (b) The fundamental TE mode. (c) The high order TE mode. (d) The fundamental TM mode (a) The 2D even TE mode distribution. (b) The odd TE mode distribution. (c) The 2D even TE mode with the amplitude distribution in z-direction. (d) The 2D odd TE mode with the amplitude distribution in z-direction (a) The relation between the mode distribution and the mismatch parameter δ (b) The odd TE mode distribution [13] (a) The normal structure of III-V/Si waveguide with 9QWs (154 nm), 50 nm SCH, 100 nm n-cladding layer and 10 nm BCB. (b) The Y-shaped structure of III-V/Si waveguide. All the epilayers are the same as the normal structure but with different InP structure. (c) The fundamental TE mode of the normal structure and its confinement factor in the active region Γ act. and Si Γ Si. (d) The fundamental TE mode of the Y-shaped structure and its confinement factor in the active region Γ act. and Si Γ Si (a) The confinement factors with different SCH thickness and BCB thickness with 100 nm n-cladding layer. (b) The confinement factors with different SCH thickness and BCB thickness with 150 nm n-cladding layer. (c) The confinement factors with different SCH thickness and BCB thickness with 200 nm n-cladding layer. (d) The confinement factors with 10 nm BCB layer, 75 nm SCH and different thickness of n-cladding layer (a) The confinement factors with different SCH thickness and BCB thickness with 100 nm n-cladding layer. (b) The confinement factors with different SCH thickness and BCB thickness with 150 nm n-cladding layer. (c) The confinement factors with different SCH thickness and BCB thickness with 200 nm n-cladding layer. (d) The confinement factors with 10 nm BCB layer, 75 nm SCH and different thickness of n-cladding layer (a) The confinement factors with different SCH thickness and BCB thickness with 100 nm n-cladding layer. (b) The confinement factors with different SCH thickness and BCB thickness with 150 nm n-cladding layer. (c) The confinement factors with different SCH thickness and BCB thickness with 200 nm n-cladding layer. (d) The confinement factors with 10 nm BCB layer, 75 nm SCH and different thickness of n-cladding layer (a) The selected structures with 75 nm SCH, 10 nm BCB, 100 nm n- cladding layer, and different QW numbers. (b) The confinement factors of different QW number cases. (c) The calculated confinement factor in a single QW with different QW numbers vii

12 4.11 The simulated cross section in the discussion of the coupling coefficient κ discussion (a), (c), and (e) are the calculated n h eff and nl eff of the fundamental mode with 400 and 220 nm height of the Si rib. (b), (d), and (f) are the calculated κ values and the confinement factors Γ act (a) The thickness of the epilayers with different n-cladding layers W nm. The cases in the comparison are composed of 75 nm SCH, 10 nm BCB, 106 nm active region (6 QWs), and with different thickness W nm of n-cladding layers. (b) The coupling coefficients κ with different n-cladding layers (a), (c), and (e) are the calculated n h eff and nl eff of the fundamental mode with 400 and 220 nm height Si rib. (b), (d), and (f) are the calculated κ values and the confinement factors Γ act (a) The thickness of the epilayers with different n-cladding layers W nm. The cases in the comparison are composed of 75 nm SCH, 10 nm BCB, 154 nm active region (9 QWs), and with different thickness W nm of n-cladding layers. (b) The coupling coefficients κ with different n-cladding layers (a), (c), and (e) are the calculated n h eff and nl eff of the fundamental mode with 400 and 220 nm height Si rib. (b), (d), and (f) are the calculated κ values and the confinement factors Γ act (a) The thickness of the epilayers with different n-cladding layers W nm. The cases in the comparison are composed of 75 nm SCH, 10 nm BCB, 202 nm active region (12 QWs), and with different thickness W nm of n- cladding layers. (b) The coupling coefficients κ with different n-cladding layers (a) The structures are composed of 75 nm SCH, 10 nm BCB, 100 nm n-cladding layer, and the different numbers of QWs. (b) The coupling coefficients κ vs. different QW numbers. (c) The coupling coefficients κl vs. the cavity length with different QW cases (a) The structures are composed of 75 nm SCH, 10 nm BCB, 100 nm n-cladding layer, and the different numbers of QWs. (b) The optical confinement factors of the total QW layers in the cases with 6, 9, and 12 QWs (a) The structures are composed of 75 nm SCH, 10 nm BCB, 100 nm n- cladding layer, and different QW numbers. (b) The optical gain curves of the 6 QW case. (c) The optical gain curves of the 9 QW case. (d) The optical gain curves of the 12 QW case. The injected carrier density of the equal amount of electrons and holes in all the cases are from to All the interval of injected carrier density between the two adjacent gain curves in all the figures are The extracted modal gain values of the different QW number cases at 1550 nm wavelength. We assume that the threshold gain is 40 cm The extracted differential gain values of the different QW number cases at 1550 nm wavelength. We assume that the threshold gain is 40 cm dg/dn N wd ww actl 5.5 The estimated values of the different QW number cases at 1550 nm wavelength. We assume that the threshold gain is 40 cm viii

13 dg/dn 5.6 The estimated N w d w W act values of the different QW number cases at 1550 L nm wavelength. The SCH and BCB layers are also changed, and all the cases are with 100 n-cladding layer. (a) 6 QWs (b) 9 QWs (c) 12 QWs. We assume that the threshold gain is 40 cm ix

14 List of Tables 3.1 Detailed material parameters Thickness of different QW numbers Detailed material parameters x

15 Abbreviations B BCB Benzocyclobutene polymer D DFB DBR Distributed feedback Distributed Bragg reflector F FP FDM Fabry-Perot finite-differnece method H hh heavy hole I IVBA Laser lh intervalence band absorption loss light amplification by stimulated emission of radiation light hole M MQW multiple quantum well Q QW quantum well S SCH separate confinement heterostructure V VCSEL vertical-cavity surface-emitting 1

16 Chapter 1 Introduction 1.1 Motivation With the advance of the digital technology, the cyber net technology brings human beings amoreconvenientlife. Thecloudcomputationallowspeopletoputalltheinformationor personal data on the network, and they are then able to process the data on the Internet at any place. Thus, people don t need to worry about the problems of data movement and storage. On the other hand, the improvement of the cybernet enables the government and private companies to manage a huge amount of information more efficiently. Although the fiber-optical communication, which is faster than traditional electrical cable line, is mature now, the electronic integrated circuit is still the core unit of the data processing. Due to the limitation of the power and bandwidth, the low processor speed and the low amount of the transmitted data in the traditional electronic circuit hinder the development of the digital technology. Besides, the large heat generation in the data center with its high density of electronic circuits enhances the global warming and might indirectly result in an ecological catastrophe. Furthermore, Moore slaw,whichpredictsthattheamount of transistors on a single chip will double in every 18 months, has met its limitation. Since the devices have been scaled down to micro-scale, the leakage current attributed to the quantum effect [14] and the thermal noise [15] dramatically decrease the device development. These drawbacks from the fundamental physics limit the development of the digital technology and the well-being of the human kind. In the past decades, the integrated photonic circuit has been proposed to replace the electronic circuit technology, where scientists have seen the end of traditional electronic devices [16]. Recently, the first micro-processor based on direct usage of light has been proposed in Nature in December 2015 [17]. The processing speed is 10 to 50 times Figure 1.1: Dispersion vs. wavelength in optical fiber [1]. 2

17 Figure 1.2: Attenuation vs. wavelength in optical fiber [2]. larger than for traditional electronic devices. This result reveals that a new era of the communication technology is arising, which means that we are no longer constrained to the limitation of Moore s law. At present, the photonic communication system is comprised of III/V semiconductor laser based signal sources and optical fibers. III/V semiconductor lasers have a good efficiency in generating the optical signal. The most popular operating wavelengths are 1300 nm and 1550 nm. Figure 1.1 shows that the dispersion at 1300 nm wavelength in glass fibers is zero, where there will be no degradation due to the wavelength dependent group velocity [1]. On the other hand, the attenuation in Fig. 1.2 reveals that 1550 nm wavelength in glass fibers has the lowest attenuation [2]. Both of wavelengths have their advantage. In the following research, we will focus on the 1550 nm laser. Traditionally, InGaAsP, a kind of quaternary semiconductor compound, is used as the optical active material, quantum well (QW), in the laser. However, the low conduction band offset of InGaAsP based devices results in a low quantum confinement of electrons and increases the leakage current. When the temperature increases, the thermionic leakage will deteriorate the leakage effect and degrade the performance of the laser [18]. Recently, the fabrication technology of AlGaInAs is improved. This kind of material has a higher conduction band offset ratio than for InGaAsP [19]. In other words, the confinement for the electrons in a AlGaInAs system is higher, which reduces the thermionic leakage current [20]. Since the effective mass of electrons is smaller than holes, the higher conduction band offset makes AlGaInAs based devices have a higher quantum confinement for electrons, and also benefits the hole carrier transport [21]. Then, the carrier distribution in the AlGaInAs system will be more uniform and thus decrease the loss induced by the high local carrier density such as nonradiative radiation, Auger recombination, and intervalence band absorption loss (IVBA) [22]. Lasers made of III/V membranes, heterogeneously integrated on SOI waveguide are expected to offer advantage for high speed modulations, but also lend themselves perfectly for integration with passive Si waveguide structures, and even co-integration with driver electronics. The main laser diodes used in optical communication are single mode dis- 3

18 tributed feedback (DFB) lasers, in which mode selectivity is obtained through a diffraction Bragg grating. In the case of III/V on Si lasers, this Bragg grating is etched in the underlying Si waveguide. For a good laser operation, sufficient optical confinements of the laser mode with both the active layer, consisting of multiple quantum wells (MQWs), and the Si grating are required [23]. In the ultimate goal, researchers want to produce an ultra high speed laser as the signal source so that we can transmit more data in the fiber per unit time. In semiconductor lasers, the modulation bandwidth is determined by the relaxation oscillation frequency fr [24], which can be expressed as: f r = 1 v g Γdg material /dn 2π en w d w W act L (I I th), (1.1.1) where v g is the group velocity, Γ is the optical confinement factor in the QW. N w, d w, W,andL are the number of QWs, the thickness of a single QW, the width of the active region, and the length of the cavity. dg material /dn is the differential gain of the material gain. With choosing a proper Γ, the whole device can operate at a low carrier density which will result in a high differential gain, and also have a good portion of the optical mode in the Si waveguide. In this thesis, we hope that the analysis can provide a good reference for engineers or researchers to produce a high quality and high speed III/V semiconductor laser. 1.2 Semiconductors Fundamental Physics of Semiconductors An optoelectronic device is basically based on a p-n junction which can provide electron and hole carriers to recombine and generate the light. In convention, the statistical physics can properly describe the carrier distribution. The Fermi-Dirac distribution describes the probability of the existence of carriers at the energy level E: f(e) = 1 1+exp[ E E F k B T ], (1.2.1) where E is the energy level of the carriers, E F is the Fermi level, k B is the Boltzmann constant, and T is the temperature. E F usually locates at the middle of the bandgap in intrinsic semiconductor materials [8]. Figure 1.3 [3] shows the typical Fermi-Dirac distribution in a semiconductor system. The density of states for different dimensions of the semiconductor structure will be different as shown in Fig. 1.4 [4]. Nevertheless, the density of states should be considered in the calculation of the carrier density. The density of states in bulk and QW materials should be expressed as Eq D(bulk) :D(E) = 1 2π (2m 2 h 2 )3/2 E, (1.2.2) 2D(QW ):D(E) = m n π h 2 θ(e E nz ), (1.2.3) 4 i=1

19 Figure 1.3: A typical Fermi-Dirac distribution in a semiconductor system [3]. Figure 1.4: (a) The density of states in the bulk semiconductor. (b) The density of states in the QW semiconductor. [4]. where h is Planck s constant, E z is the quantized energy in quantum wells of z direction. θ(e E nz ) is the Heavside function [8]. m is the effective mass of the carriers. Figure 1.3 shows that the density of states in the bulk material is a energy related parabolic function; however, the density of states in QW system is as a Heaviside function of the energy [4]. Hence, the carrier density can be clearly stated as the product of the Fermi- 5

20 Figure 1.5: (a) The Fermi-Dirac distribution in a n-type semiconductor system. (b) The Fermi-Dirac distribution in a p-type semiconductor system. [3] Dirac distribution and density of states in Eq n(e) = D c f(e) p(e) = D v [1 f(e)], (1.2.4) where n(e) andp(e) areelectronandholecarrierdensityatenergylevele, andd c and D v is the density of state of electron and hole carriers. In practice, we need to increase the amount of electrons or holes for the devices. So we add the impurity in the semiconductor material. For n-type material, we add donor-like impurity in the material. In contrast, for p-type material, we add acceptor-like impurity. Thus, the Fermi energy will not be located at the mid-gap. Figure 1.5 shows the Fermi- Dirac distribution of the n-type and p-type material [3]. In the n-type material, the electron density will be far higher than the hole density due to the upward shift of the Fermi level. In p-type system, it works on the same principle, and the hole carriers will dominate in the p-type system Energy Band in a Typical QW Device Figure 1.6 shows the energy band in a typical QW based optoelectronic device at 0V. The device is composed of three parts, which are n-type, intrinsic doping, and p type region. Normally, the Fermi level will be above the conduction band edge in the n-type region and below the valence band edge in the p-type region. Thus, there will be lots of electron and hole carriers in n-type and p-type region. When we apply the bias, carriers can be injected into the intrinsic region. The main function of optoelectronic devices is to utilize the emission light from electron-hole pairs radiative recombination for other applications. In order to reach the maximal light generation, scientists add the QW epilayer into the device. Thus, the large amount of electrons and holes will be trapped in the QW which is the local energy minimal region. LEDs and laser devices are mainly based on the spontaneous radiative emission and stimulated radiative emission respectively. It is noticed that the energy state distribution in QW materials is different from the bulk materials. 6

21 Figure 1.6: The scheme of a p-n QW device. Figure 1.7 shows the concept of the energy state distribution in a QW along the z direction. From Schrödinger Equation [8], the energy states will be discrete in the direction of the energy confinement. The energy of carriers will be in the following form: E n = h2 2m z k 2 n,z = h2 ( nπ ), 2m z d z E t (k x,k y ) = E n + h2 2m t (k 2 x + k 2 y), (1.2.5) where n is the integer coefficient. d z is the width of the quantum well along the z-direction. k x, k y,andk z are the wave vector of x, y, andz. m z is the effective mass of the z-direction, and m t is the effective mass in the x, y directions. Carriers can only transit between these states. Besides, the emission energy will no longer be the bulk bandgap. The new effective bandgap will be the energy difference of the ground states of the electrons and holes, and this determines the spectrum of the emission light. In the semiconductor materials, not only the radiative recombination exists but also some undesired loss mechanisms. Figure 1.8 shows the different carrier behaviors in a normal optoelectronic device. SRH is the defect assisted non-radiative recombination [25]. Auger recombination is a three particles involved mechanism with the phonon assistance [26], where the recombined energy will excite the other carrier to higher energy state. I leakage is the leakage current. As we know, the electron has a lighter effective mass so that electrons will have a higher chance to fly over the QW due to its overly high mobility. In the past, the InGaAsP based 1550 nm laser has suffered from the leakage current, and the performance is not so good [18]. However, the high conduction band offset ratio of AlGaInAs might replace InGaAsP in 1550 nm laser and mitigate the leakage issue. 7

22 Figure 1.7: The illustration of the quantized energy states in the QW. 1.3 III-V/Si Hybrid Waveguide The laser device used in the integrated photonics circuit is basically a kind of hybrid waveguide. We use different kinds of the material to make the optical field concentrate in the active region which will increase the power of the laser, and thus produce a high coherence and high power density of the optical field. Besides, researchers might use photonic crystal or Si waveguide with some techniques to guide the signal generated by the laser source out, and do further modulations. Before we talk about the detail of the whole story, we should first know what is a waveguide. Figure 1.9 shows how the light travels in a hetero-material system. n f and n s are the refractive index corresponding to the epilayer, n f is larger than n s.fromthesnell slaw shown below: n s sinθ s = n f sinθ f, θ f < sin 1 ( n s n f ), (1.3.1) where sin 1 ( n s n f )isthecriticalangelθ c. As the ratio of n s n f decreases, the possibility for the light confinement in the middle layer becomes higher [5]. In practice, the active region is the higher refractive index layer. As a result, increasing the refractive index might increase the optical confinement factor of the optical field in the active region, and thus introduce a higher gain and higher differential gain. From Maxwell s equations and solving the boundary condition of the waveguide, there will be two different types of the fields which are TE and TM [5] in the waveguide. The TE and TM field are referred to as the electric and magnetic field normal to the incident plane respectively as shown in Fig Besides, there will be different orders of TE or TM modes existing in the system, when they meet the guiding condition in the waveguide. Figure 1.11 shows the different orders of TE modes that exist in a planar waveguide [5]. Here, we don t discuss the detailed 8

23 Figure 1.8: (a) The illustration of carrier behavior in a optoelectronic device. (b) The scheme of Auger recombination. theory. The detailed theory applied in our calculation will be introduced in the further chapter. Figure 1.12 shows a typical III-V/Si hybrid waveguide. The laser signal will be generated 9

24 Figure 1.9: The concept of the light travels in a heteromaterial system [5]. (n f >n s ) Figure 1.10: The scheme of the TE and TM filed propagate in a planar waveguide [5]. Figure 1.11: Different orders of TE mode in a planar waveguide [5]. in the III/V layer, and coupled out by using the taper like Si waveguide. Figure 1.13 from Ref. [7] shows that the confinement of the Si region will increase with increasing the width of the Si layer. That is how the III/V laser couples out the signal combined with 10

25 Figure 1.12: A typical III-V/Si hybrid waveguide [6]. Figure 1.13: The calculation of the confinement factor for a even fundamental mode of the III/V active region and the Si from the group from California Institute of Technology [7]. a Si based taper passive waveguide. When the signal is near the taper region, the energy will gradually transfer to the Si waveguide due to the increasing width of the Si layer and the decreasing width of the III/V layer; namely, couple out the device. 11

26 Figure 1.14: The Gain spectrum of unstrained 8 nm GaAs/Al 0.15 Ga 0.85 As QW/QB with different injection carrier density (10 18 cm 3 )[8]. 1.4 Semiconductor Laser Optical Gain Laser is the abbreviation of light amplification by stimulated emission of radiation [27]. By the definition, the optical power will be enhanced due to the stimulated emission of radiation, where we use the concept of optical gain to quantify the level of the optical enhancement. The unit of the optical material gain, g material,iscm 1,whichmeans that the optical gain refers to how the optical power will be enhanced per 1 cm propagation. Figure 1.14 shows an example of the gain spectrum and quantized energy levels of a semiconductor QW device which is unstrained GaAs/Al 0.15 Ga 0.85 As QW/QB with 8nmthickness[8]. Wecanseethatthedifferentpolarizationsoftheemissionlightcan contribute to the gain spectrum in an unstrained QW. Besides, there are different peaks existing in both spectra, which are attributed to the different band to band transitions. Furthermore, carriers might have some interactions with other carriers and phonons, and these will make the gain spectra broader. In the last paragraph, the case I mentioned is with unstrained QW. However, in the real device, QWs might suffer from different degree of strain due to the lattice mismatch between the substrate and QWs. The different kind of strain will change the relative position of the heavy hole and light hole valence bands as shown in Fig [8]. For the compressive strain, the heavy hole valence bands and light hole valence bands will separate. Then, the heavy hole valence bands will move upward, and the light hole valence bands will move downward. In contrast, the tensile strain will result in an opposite trend of the compressive strain. By the Fermi-Dirac distribution, we know that carriers near the bandedge or lowest band will dominate the band to band transition. Consequently, the conduction band to heavy hole band transition will dominate in compressive strain QWs, and the conduction band to light hole transition will dominate in tensile strain QWs. From Ref. [8], TE and TM polarization of the light emission will dominate in the conduction band to heavy and light hole transition individually. Figure 1.16 shows the results of the emission spectrum influenced by different kinds of strain [8]. 12

27 Figure 1.15: The band edge of compressive, unstrained, and tensile 8 nm GaAs/Al 0.15 Ga 0.85 As [8]. Figure 1.16: The band edge of compressive, unstrained, and tensile 8 nm GaAs/Al 0.15 Ga 0.85 As. [8] Besides, we should consider the optical confinement factor, Γ, of the QW region. Thus, the optical gain should be expressed as Γg material which we call the modal gain. 13

28 Figure 1.17: The scheme of an optical cavity. R 1 and R 2 are the reflective index of the mirrors. Γg material and α are the optical gain and loss of the internal cavity, and the unit of both parameters is cm 1.Listhelengthofthecavity Principle of Semiconductor Lasers In semiconductor laser devices, we inject carriers into QWs. Then some of the carriers spontaneously emit the light and further stimulate other carriers. Then, we need a cavity to preserve the light to excite more carriers, and increase the stimulated emission so that we can amplify the optical power in laser devices. Figure 1.17 shows a Fabry-Perot (FP) cavity with the optical gain material. The optical power after a round trip can be expressed as follows: P = P 0 R 1 R 2 e 2(Γg material α)l, (1.4.1) where P 0 is the initial power. When the lasing starts, the gain, Γg material,willbemuch more than than the mirror loss, R 1 and R 2,andtheinternalloss,α. When the optical power is enhanced by the stimulated emission light, more carriers will recombine. Then the power will decrease with carrier recombination. But the injected electronic current will supply the consumed carriers, the optical gain will increase again. After a while, the laser system will reach the steady state in a certain degree of population inversion [28]. Normally, the modal gain, which is calculated by optical confinement factor and material gain, of the steady state, g th,willequalallthelossmechanismsasfollows: 1 = R 1 R 2 e 2(g th α)l, g th = α + 1 2L ln 1 R 1 R 2, (1.4.2) where g th is the threshold gain. Apart from the Fabry-Perot cavity which uses the mirrors for the reflectivity, some lasers also use the grating to induce the reflection of the optical field. As Fig shows, when the optical field reaches the Bragg reflective condition which is relating to the parameter Λ, the optical field will be reflected. 14

29 Figure 1.18: The scheme of a grating. n A and n B are different refrative index of the material. Λ is the period of the grating [5]. Figure 1.19: Four typical optcal cavities applied in semiconductor laser. (a) Fabry-Perot Laser. (b) DBR laser. (c) DFB laser. (d) VCSEL. [9, 10] Figure 1.19 (b) and (c) show the typical structures of distributed Bragg reflector lasers (DBR lasers) and distributed feedback lasers (DFB lasers) [9]. The grating is on the two sides of the main active regions in DBR lasers, and the active region is deposited on top of or below the grating section in DFB lasers DFB Lasers Compared to other kinds of lasers, semiconductor lasers are lighter, more efficient, and have smaller volume and a longer lifetime. As a result, the semiconductor lasers have drawn more and more attention, and are considered as a light source for many applications (e.g., optical communication, photonics integrated circuits, etc). The two key components 15

30 Figure 1.20: The scheme of a diffraction grating. [11] of semiconductor lasers are materials and cavities. For materials, we need to choose the semiconductor compounds with a proper bandgap as the active materials. Besides, the design of epilayer is also important to the optical field distribution and the carrier injection, where these parameters influence the lasing efficiency. The other important factor in semiconductor lasers is the cavity. Figure 1.19 shows four typical kinds of laser cavities in semiconductor lasers. The first one is the Fabry-Perot (FP) cavity. The cavity in FP lasers is comprised of two mirrors sandwiching the semiconductor epilayers. The advantage of FP lasers is their easy fabrication. However, FP lasers are usually under multiple longitudinal mode operation which might result in a larger dispersion [29] in optical fibers. Thus, some scientists have proposed the grating as a advanced filter of the longitudinal mode [30 32]. In the previous section, we have briefly introduced how DBR and DFB lasers (in Fig (b) and (c)) work and their structures. With the grating based laser devices, lasers can be easily operated under single mode. The fiber dispersion will be lower than for multimode lasers, which is beneficial for the transmission in optical fibers. Although DFB and DBR lasers are both based on a grating, they however have a different behavior, especially the temperature behavior. Due to the nature of the structure, the grating part will experience the similar temperature variation as the active region in DFB lasers. DBR lasers in contrast will suffer from the loss and mode-hopping more due to the inconsistent condition between the active region and grating [9]. With the mature development of the fabrication technology, the DFB laser is consequently a promising device being the backbone of the photonics integrated circuits. Another type of grating based lasers is the vertical-cavity surface-emitting laser (VCSEL) in Fig (d). The longitudinal modes in VCSEL are vertical, which are different from traditional FP, DBR, and DFB lasers. The VCSEL has lower cost in the fabrication process, because it can be deposited on the substrate in 2D array, which is beneficial for mass production. Furthermore, the VCSEL also has a better carrier confinement in the active region and a lower divergence angle of the output light [33]. However, the difficulty in producing the VCSEL and the limited output power has limited its development. In grating theory, we assume that the refractive index in a transparent diffraction grating 16

31 is a sinusoidal function as in Fig. 1.20: n r (z) = n r0 + n r1 cos(2β 0 z +Ω), (1.4.3) β 0 = π Λ, where n r0 n r1.ωisthephaseshiftatz =0. Thefollowingequationswillbebasedon the coupled wave theory [32]. We can ignore the higher order terms and the wavevector k, propagatingindiffractivegratingshouldbeexpressedas: k(z) 2 = β 2 +4k 0 n r0 κcos(2β 0 z +Ω), (1.4.4) β = k 2 0n r0, κ = πn r1 λ 0, where λ 0 is the propagation wavelength. Then, we replace Eq into the differential equation of electric wave: d 2 E dz 2 +[β2 +4βκcos(2β 0 z +Ω)]E =0. (1.4.5) We can sperate the solution into forward and backward propagating waves: E(z) = E + (z)+e (z) E + (z) = R + (z)e iβ 0z E (z) = R (z)e iβ 0z, (1.4.6) where E(z), E + (z), and E (z) arethetotalelectricfield,forwardpropagatingelectric field, and backward propagating electric field. R + and R are the amplitude of E + (z), and E (z). We then substitute Eq into Eq We can obtain: dr+ dz iϱr+ = iκr e iω dr dz iϱr = iκr + e iω (1.4.7) ϱ = β2 β 2 0 2β 0 β β 0, As the wavelength is the Bragg wavelength, λ B,δ will be zero. The imaginary part in Eq will diminish, and λ B will dominate the propagation wavelength. Note that the coupling coefficient, κ, standsforthereflectivitypercm of coupling field in the grating from R + to R or from R + to R,andisanimportantparameterinthecoupled-wave theory. After solving the whole system analytically, we obtain that the power reflectivity at λ B, R, isasfollows[5]: R = tanh 2 (κl), (1.4.8) where L is the length of the device. Equation reveals that κl will have a decisive position of the reflectivity in the grating. κl should be properly chosen. Otherwise, the output power will be too low with overly high values, or the reflectivity will be too low to confine the optical field with overly low values. Normally, κl values in DFB lasers will be 1 3. At first glance, increasing κ values might help us to decrease the device length, which is beneficial for minimizing volumes of devices. However, the overly short devices are deficient for the heat dissipation, and the imperfectly cleaved facets on two sides of lasers might result in a larger scattering loss. 17

32 1.5 Thesis Overview In the thesis, our main purpose is to provide a systematic numerical analysis for the design of III-V Si hybrid high speed 1550 nm lasers. To make it easier to read the content, we have illustrated the necessary background of the physics of 1550 nm hybrid lasers in Chapter 1. Then, how the simulation models work and the physics behind them is explained in Chapter 2. As I mentioned in the previous section, AlGaInAs is a more suitable material for producing 1550 nm high speed lasers. Chapter 3 shows the calculated energy band of InGaAsP and AlGaInAs based QWs. The result clearly reveals that it is worthy to substitute the main stream material of the 1550 nm emission QWs, InGaAsP, by AlGaInAs. As the QW material was chosen, we then observe how the optical fields distribute in the III-V/Si hybrid waveguide in Chapter 4. With the help of commercial software, we can easily access the mode distribution, effective refractive index, and confinement factor. We vary the compositions including QW numbers and different thickness of epilayers in the III-V/Si hybrid waveguide, and observe the performances. The results point out that there will be some trade-off in choosing different epilayers. We extract the proper confinement factors with different QW numbers for the further calculation. In Chapter 5, the modal gain spectra with different QW numbers are calculated with the gain calculator. We further estimate the gain curves with different injected carrier density in different QW number case. The differential gain, a physical indicator of the laser operation speed, is also shown in this chapter. A comparison of different QW numbers is plotted. Lastly, we summarize the results of the whole thesis, and give the future perspective in Chapter 6. 18

33 Chapter 2 Methodology In this Chapter, we introduce the models which are applied in this thesis. At first, we use the commercial software SimuLase [34 36] to calculate the band structure, the wavefunctions and the energy states of QWs. The wavefunctions and energy states are calculated by the k p method [35, 36] which is one kind of perturbation theory for the Schrödinger equation. On the other hand, the commercial software Fimmwave [37] which is an optical mode solver gives us the optical performance of different devices. The confinement factors and other physical parameters of the III-V/Si hybrid waveguides are analyzed under properly implanted parameters. After finishing the energy states and confinement factor calculations, we then substitute these physical quantities into the optical gain calculator which is developed by Prof. Lysak [38]. The optical gain curves and differential gain with different structures are obtained, where these parameters are important physical indicators of the laser performance. 2.1 Energy Band and Quantum Level The III/V laser systems are comprised of different III/V compounds. However, the energy bandgap of the semiconductor compounds is hard to model for alloy compositions. Fortunately, some scientists proposed the analytical equations with bowing terms to express the relation between the alloy composition and the energy bandgap. The bandgap of the materials applied in the thesis is in the following form [19]: E g (Al x Ga y In 1 x y As) = x 0.629y x y xy 2.0xy(1 x y), (2.1.1) E g (In x Ga 1 x As y P 1 y ) = (1 x) 1.110y (1 x) y (1 x)y 0.304(1 x) 2 y (1 x)y 2. (2.1.2) The band offset ratio of the heteromaterial system is according to Ref. [39]. For the Schrödinger equation, the eigenvalues and eigenvectors can be solved as follows: ( p2 + V )ψ(r) =Eψ(r), (2.1.3) 2m where p is the momentum operator, m is the mass of the carriers, V is the potential, E is the eigenenergy, and ψ(r) isthesolutioninthreedimensions(3d).sincethecrystalline semiconductor materials are highly symmetric to the Brillouin zone, the wave function can be expressed in a periodic function by Bloch s theorem: ψ n,k (r) =e i k r u n,k (r),r (2.1.4) where ψ n,k is the wavefunction with wavevector k under band integer index n. u n,k (r) is the periodic function. Then, the Schrödinger equation can be expressed under the 19

34 perturbation theory in the k p form: H t ku n,k (r) = H 0 + H k = E n,k u n,k (r), (2.1.5) H 0 = p2 + V, 2m (2.1.6) H k = h h (k p)+ [ V (p + hk)] σ, m 4m 2 c2 (2.1.7) where H 0, H k,andhk t are the Hamiltonian without perturbation, the perturbation term, and the total Hamiltonian; σ is a Pauli matrix. Then, the expressions of the results are[40]: u n,k = u n,0 + h m n n E n,k = E n,0 + h2 k 2 2m + h2 m 2 <u n,0 k p u n,0 > u n E n,0 E,0, (2.1.8) n,0 n n <u n,0 k p u n,0 > 2, (2.1.9) E n,0 E n,0 Besides, the strain induced by incompatible lattice constants in the heteromaterial system has been considered, all the detailed calculations can be found in Ref. [41]. In order to estimate the carrier distribution in the quantum well, the Poisson and driftdiffusion equations are used in the software: (ϵ V (r)) = n(r) p(r)+n A (r) N D (r), (2.1.10) J n (r) =qµ n n(r) V (r)+qd n n(r), (2.1.11) J p (r) =qµ p p(r) V (r) qd p p(r), (2.1.12) where V is the potential, and ϵ is the static dielectric constant. N A and N + D are the fixed charge of ionized doping density. n and p are the free carrier concentration of the electron and hole. q is electron charge C. µ n,p are electron and hole mobility. D n,p are the coefficients of diffusion, and J n,p (r) are the electron and hole current, respectively. 2.2 Optical Mode Calculation In general, the optical mode solvers with different numerical methods are solving Maxwell equations [42]: E = B t, (2.2.1) B = D + J, t (2.2.2) D = ρ, (2.2.3) B = 0, (2.2.4) where E, and B are the electric and magnetic fields, D and H are the electric and magnetic flux density, J is the current density, and ρ is the charge density. In our case, the optical field is calculated in a passive waveguide. Therefore, J and ρ are considered 20

35 Figure 2.1: The concept of the finite difference method [12]. as 0. Under the boundary conditions, the optical fields can be solved in the waveguide with various materials. However, it is impossible to use simple analytical formulas to model the solutions. Luckily, Photon Design [37] have developed a series of reliable and versatile mode solvers, Fimmwave. The software provides several different numerical solvers which can solve any kind of geometric structure with different dielectric materials. Finite-differnece method (FDM), which is a numerical method, is a suitable solver to solve the optical modes in our case due to its fast computation and accuracy [43]. Basically, Maxwell equations are differential equations. Thus, the FDM is proposed to solve the differential equation with some numerical techniques. The first order of the differential is expressed as [12]: f f(x + h) f(x) (x) =lim. (2.2.5) h 0 h As Fig. 2.1 shows, we can make a approximation with the FDM as follows: f (x) f(x + h) f(x). (2.2.6) h After the mode calculation is done, the confinement factor which is very important for optical gain calculations can be achieved. The confinement factor Γ is estimated under the following equations: Γ= ϵ S 0c n(x, y)e 2 (x, y)dxdy, (2.2.7) 4 P total where S is the specific chosen block, ϵ 0 is the dielectric permittivity in vacuum space, c is the speed of light, E(x, y) andn(x, y) arethepositiondependentelectricfieldand refractive index profile, and P total is the total power of the fields in space. Furthermore, the software provides the fraction of TE field in each solved mode. The value of the TE fraction provides us another reference to choose the mode. The TE field is polarized in x-direction, and the TM field is polarized in y-direction. It is estimated by 21

36 the following : T Efraction = The effective index n eff can estimated in the following: E x H y dxdy P total. (2.2.8) β j = k z, (2.2.9) n eff = 2πβ j λ 0, (2.2.10) where β j is the propagation constant of mode j, andλ 0 is the wavelength of the optical fields. With the help of n eff,thecouplingcoefficientκ in the square grating profile can be approximately estimated by using the following equation [44]: κ = 2(nh eff nl eff ) λ 0 sin(m π l h ), (2.2.11) Λ where n h eff and nl eff are the effective refractive index in the unetched region (higher effective index) and etched region (lower effective index) respectively. λ 0 is the propagation wavelength, m is the order of the grating, l h Λ is the duty cycle of the grating. 2.3 Optical Gain The optical gain calculator we used is developed by Prof. Lysak [38]. For achieving a higher accuracy calculation, we inserted the energy levels by considering the strain effect in SimLase and optical confinement factors in Fimmwave into the gain calculator. The gain calculator is based on the following equation: πq 2 h qw (n qw,p qw,t) = Γ m 2 0ϵ 0 n eff ce tr g pol E cv (g pol c hh + gpol c lh )de eh, (2.3.1) g pol c yy(n qw,p qw,t) = ρ c yy (E eh )µ 2 pol,c yyπl(e eh,e tr ) (f c (E eh,n qw,t) f yy (E eh,p qw,t)), (2.3.2) (2.3.3) where gqw pol is the total modal gain with TE or TM polarization in the QW, n qw and p qw is the carrier density (cm 3 ), Γ is the inserted confinement factor; q is electron charge C, h is the Plank s constant, ϵ 0 is the permittivity of vacuum, m 0 is the mass of free electrons, c is the speed of light, n eff is the effective refractive index; E tr is the photon energy, E eh is the transition energy between electron and hole states, E cv is the lowest transition energy of the bulk bandgap, label yy refers to heavy hole (hh) and light hole (lh) state, respectively; ρ c yy is the density of allowed transition states, µ 2 pol,c yy is the transition matrix for different transitions and polarization, L(E eh,e tr )isthelinewidth enhancement factor, and f c and f yy refers to occupation factors of electrons and holes. Noticed that the carrier density can be set up manually, and it assumes that the carrier distribution is uniform in the QWs. 22

37 The occupation factors in the program are : f c (E,n qw,t) = [1+exp( E E fc(n qw,t)) )] 1, k B T (2.3.4) f yy (E,p qw,t) = [1+exp( E E fyy(p qw,t) )] 1, k B T (2.3.5) (2.3.6) where E fc and E fyy are the Fermi energy of electrons and holes, and k B are Boltzmann constant. The Fermi levels, E fc and E fyy,canbeobtainedintheinversionofthecarrier density, n qw and p qw, in the QWs: n qw = Σ i D c (Ec)ln[1 i + exp( E fc Ec i )], (2.3.7) K B T p qw = Σ i D yy (Eyy)ln[1 i + exp( Ei yy E fyy )], (2.3.8) K B T (2.3.9) where the integer i represents the different energy state. In convention, the density of states is expressed as Eq However, the case in Eq is the infinite quantum well, the energy levels will be shifted in the finite quantum well, and the density of states will be changed slighly. Here, we assume that the amplitude of the 2D density of states is approximately equal to the 3D density of states at a confined energy level with finite QW in the z-direction. The density states of electrons and holes D i c and D i yy in subband i are : Dc(E i c) i = ( 2m c 1 h 2 )1.5 E i 4π 2 c, (2.3.10) Dy(E i yy) i = ( 2m yy h 2 ) E 4π yy, i (2.3.11) 2 (2.3.12) where E i c and E i yy are the quantized energy level. On the other hand, the density of allowed transition states ρ c yy is: ρ c yy =Σ i ( 2m c yy h 2 ) π 2 E i c,yy,h(e E i c,yy), (2.3.13) where E n c,yy is the different energy transitions between the quantum levels of electrons and holes, H(E E n c,yy) is the Heaviside function, and m c yy is the reduced mass: 1 m c yy = 1 m c + 1, (2.3.14) m yy where m c and m yy are the effective mass of electrons and holes. Besides, the transition rule should be considered in the optical gain calculation. The energy state transition of carriers should obey the Fermi golden rule, momentum conservation, and energy conservation. The transition strength is related to the inner product of wavevector k and the optical field E.Thetransitionstrengthelementsofdifferentpolarizationsandstatetransitions 23

38 Figure 2.2: The Gain spectra of 6 nm single Al Ga In 0.7 As/Al 0.23 Ga In As QW/QB with cm 3 injected electron and hole density under different relaxation times. are: µ 2 TE,c hh = 3 4 (1 + cos2 θ)m 2 b, (2.3.15) µ 2 TE,c lh = 1 4 (5 3cos2 θ)m 2 b, (2.3.16) µ 2 TM,c hh = 3 2 (1 cos2 θ)m 2 b, (2.3.17) µ 2 TM,c hh = 1 2 (1 + 3cos2 θ)m 2 b, (2.3.18) (2.3.19) where Mb 2 is the momentum matrix element which varies for different semiconductors, and it is expressed: Mb 2 = m 0 6 E p, (2.3.20) where E p is the energy parameter that depends on the material. The equations above are good to describe the carrier dependence of the optical gain. The energy states in semiconductor materials, however, are not that ideal as the theoretical assumption. Carriers might have some interactions with other carriers or phonons in the real devices, and there will be some substates existing in the energy diagram along the confinement direction. The emission spectrum is not that sharp as the ideal case. 24

39 As a consequence, the line width broadening factor L(E eh,e tr )isutilizedtomodelthe interaction of subband transitions. Here, we used the Lorentzian line shape function to model the line broadening factor: L(E eh,e tr ) = 1 2π Γ s, (2.3.21) (E eh E tr ) Γ 2 s Γ s = h τ s, (2.3.22) where Γ s is the average broadening function of the electron and hole states, and τ s is the relaxation time of electrons and holes. Figure 2.2 clearly shows how the relaxation times τ s influence the shape of the gain spectra. Increasing the relaxation time means that the carriers need a longer time to transit between subbstates. In contrast, decreasing the relaxation time will result in a easier transition between substates of carriers, and thus makes the spectrum broader. Normally, a τ s of 0.1ps is suitable and commonly used in most cases in the simulation [8]. 25

40 Chapter 3 Energy Band Analysis In this chapter, we will give some evidence that AlGaInAs has a strong potential to replace InGaAsP in the 1550 nm lasers by analyzing the energy band diagram of QW/QBs. The comparison between both materials of the energy band, band-off set ratio, strain, carrier distribution are drawn in the discussion. In order to make a fair comparison of different materials, we construct a simple structure in the analysis in this chapter. Figure 3.1 shows that the structures are comprised of single 6 nm QW sandwiched by two 10 nm QBs with 40 nm n-type ( cm 3 )InPand40nmp-type( cm 3 )InP.Thecompositions of QW/QB are Al Ga In 0.70 As /Al 0.23 Ga In As in the AlGaInAs case, and In Ga As 0.79 P 0.21 /In 0.71 Ga 0.29 As P in the InGaAsP case. Both the bandgap of QWs and QBs in AlGaInAs and InGaAsP are the same. Traditionally, InP or Si are regarded as the candidate substrate in the photonics devices. In spite of the high cost of InP, the InP based optical devices in 1550 nm photonics integrated circuits have a better and more reliable performance due to the lattice mismatch issue. Here, we therefore consider the InP as the substrate in our structures. In the simulation, we assume all the materials are under 300 K. The material parameters are in Table 3.1. The interpolation method estimating the parameters in different alloy compositions is in Ref. [19]. The mobility of electrons and holes in the program is according to Ref. [45]. 3.1 Energy Band Diagram and Strain Distribution Figure 3.2 shows the band diagram and quantum levels of both the AlGaInAs and In- GaAsP based devices. The blue dash line is the bulk conduction band edge. The orange Figure 3.1: The simulated single QW and two QBs structures with different material systems. 26

41 Table 3.1: Detailed material parameters binary material GaAs AlAs InAs GaP InP E g (ev) (ev) m e /m γ γ γ a 0 (Å) a (ev) b(ev) C 11 (10 11 dyn/cm 2 ) C 12 (10 11 dyn/cm 2 ) ϵ r E g is the bulk energy bandgap. 0 is the energy of the spin-orbit splitting. m e /m 0 is the relative effective electron mass to bare electron mass. γ 1,2,3 are Luttinger parameters which are utilized to estimate the effective mass of holes. a 0 is the lattice constant. a and b are the deformation potential. C 11 and C 12 are strain constant. ϵ r is the relative dielectric constant of the materials. dot line is the heavy hole band, and the yellow solid line is the light hole band. The sold lines with hollow circles are the different quantized energy states along the z-direction. Here, we only show the energy diagram in the active region. Due to the lattice mismatch, the each of the epilayer suffers from the strain induced by lattice mismatch. As I mentioned in Chapter 2, the strain separates the valence band into different bands which are the heavy hole band hh and light hole band lh. Apparently, the barriers in both of the cases suffer from tensile strain, and the well suffers from compressive strain. With the valence band separation, the bandgap of the barriers should be considered as the difference 27

42 Figure 3.2: Band diagram and quantum level of AlGaInAs and InGaAsP based devices. between the conduction band and light hole band in the strained semiconductors. Again, to make the comparison fair, we made the energy bandgap of AlGaInAs and InGaAsP almost the same, where the bandgap of the AlGaInAs and InGaAsP barriers are ev and ev respectively. The slight difference between the two structures does not influence the conclusion. On the other hand, because the thickness of the quantum well is close to the order of de Broglie wavelength of electrons and holes, the energy distribution along the z-direction is discrete as I mentioned in Chapter 2. It is noticed that the carriers are still free in x-y plane. As a result, the effective bandgap in the quantum well will be the energy difference between the ground states of electrons and holes. Here, the effective bandgap of AlGaInAs and InGaAsP are ev and ev, which are almost the same. These values are corresponding to the 1575 nm wavelength. From the Fermi-Dirac distribution, we know that the occupation factors will decrease, while the energy is away from the band edge. Thus, we are only concerned about the lower energy states. After considering the strain induced band separation effect and the quantized energy levels in the QW region along z-direction, we put both amended energy band diagrams together and make a comparison as shown in Fig The blue dash line is the effective conduction band of AlGaInAs, and the red dot line is the effective conduction band of InGaAsP. The yellow solid line is the effective valence band of AlGaInAs, and the purple dash dot line is the effective valence band of InGaAsP. The conduction band offset of the AlGaInAs and InGaAsP case are ev and ev, and the valence band offset of the AlGaInAs and InGaAsP case are and We found that the AlGaInAs based device has a larger conduction band offset ratio. The conduction band offset ratio of AlGaInAs and InGaAsP are and There will be a better confinement of electron carriers in the AlGaInAs case. Since the electrons have a much higher mobility than holes, the leakage problem is mainly caused by electrons. Hence, the better electron confinement 28

43 Figure 3.3: Effective band diagram and the band offset of AlGaInAs and InGaAsP based devices. can reduce the electron leakage and thermionic emission current effect. This means that the AlGaInAs based device will have a better efficiency and temperature performance. According to Fig. 3.4, both cases of the barriers suffer from the tensile strain, and the wells suffer from compressive strain, where the light hole band is dominant in the barrier region and the heavy hole band is dominant in the well region. Due to the low hole mobility, the holes can not spread well among the MQW case. This is very deficient for high speed DFB lasers with large number of QWs. In InGaAsP devices, the large valence band offset make the situation worse. If we can reduce the valence band offset in a proper amount, carriers might have a higher chance to propagate along the barriers, and the light hole band makes carriers move faster. Then, the carrier spreading can be improved. With amoreuniformcarrierdistribution,thedifferentialgainoflaserswillbehighercompared to the non-uniform case under the same power. 3.2 Free Carrier Distribution To further observe how the valence band ratio would influence the transport, we increase the number of QW to six in both of the cases with the same parameters. Figure 3.5 shows the free electron and hole carrier distributions ρ n and ρ p of two material systems. The injected total carriers are the same, where the AlGaInAs and InGaAsP case are under 29

44 Figure 3.4: The strain distribution along the z-direction of AlGaInAs and InGaAsP based devices. external driving voltage 0.79 V and V. The relatively high carrier density regions are the QWs. Noticed that there are some abnormal peaks at the boundary between the QW/QB interface due to the numerical problem. We will first focus on the hole distribution. According to the Poisson and drift-diffusion equations, the holes with lower 30

45 Figure 3.5: The free carrier distribution along the z-direction of AlGaInAs and InGaAsP based devices. mobility has more impact on the steady state solutions in the transport system than electrons. Fig. 3.5 (b) shows that the AlGaInAs system has a more uniform hole carrier distribution than InGaAsP system. The low effective valence band offset of the AlGaInAs system improves the hole carrier distribution among the active region. Finally, the steady 31

46 state results show that more carriers in the AlGaInAs device can reach the last QW which is close to the n-inp. With the Coulomb interaction, the more uniform hole distribution results in a more uniform electron distribution. Hence, our results show that the AlGaInAs can provide a better circumstance for the carrier injection. The better carrier distribution results in a more average carrier density in each QW, and an increased differential gain. In more non-uniform carrier distribution case with the same output power, some of the QWs will operate at high carrier density, and it will reduce the differential gain which determines the relaxation oscillation frequency and the modulation bandwidth. We further increase the Al composition in the barriers in the AlGaInAs cases. The increased composition of Al will result in a higher bandgap and larger conduction and valence band offset as shown in Fig The heavy and light hole band separation, and the quantum confined energy in the QWs have already been considered into the band calculations. Figure 3.7 shows the calculated carrier distributions with the same total carrier density in both cases. The driving voltage of the barriers with Al 0.23 Ga In As and Al 0.25 Ga In As devices are 0.79 V and V. You can clearly see that the elevated energy bandgap of the barriers makes carriers easier to be confined in the deep well of the first QW which is near the p-inp region. Thus, the carrier distribution of the barriers with higher bandgap makes the carrier distribution more non-uniform among the active region. Although reducing the barrier height can improve the carrier distribution, the insufficient bandgap of the barriers might result in a lower confinement for carriers in QWs. Consequently, the temperature dependent performance will degrade, which will perform similarly to the InGaAsP based devices. 32

47 Figure 3.6: The energy band diagram along the z-direction of the AlGaInAs device with different barrier compositions Al 0.23 Ga In As and Al 0.25 Ga In As. 33

48 Figure 3.7: The free carrier distribution along the z-direction of the AlGaInAs device with different barrier compositon Al 0.23 Ga In As and Al 0.25 Ga In As. 34

49 Chapter 4 Study of Optical Field Distribution In this chapter, we are going to introduce the second important physical property of the laser system, which is related to the optical field. In laser systems, carriers recombine, emit the light, and stimulate other carriers to recombine. This cycle keeps going, and the system will reach the steady state. How the generated light distributes so that devices can work more efficiently is the main topic in this chapter. Here, we used the software Fimmwave to estimate the pure optical field distribution in the passive waveguide which is comprised of the III/V waveguide with the active region and the Si waveguide. Because the DFB laser is a complicated structure, we only simulate the cross section shown in Fig. 4.1, which is enough for us to evaluate the performance. The optical confinement factor in the simulated cross section will be used to estimate the optical gain in next chapter. Our simulation structure is shown in Fig All the materials of the epilayers and their refractive index are listed in the figure. W is the thickness of the n-cladding layer. X is the thickness of separate confinement heterostructure (SCH) layer. Y is the thickness of the active region (MQWs). Z is the thickness of the Benzocyclobutene polymer (BCB) which is the material for the bonding layer. We will base on the structure in Fig. 4.2, and change the shape, W, X, Y,andZ value in the simulation. How the optical confinement factor and the effective refractive index are influenced by these parameters will be presented. The optical confinement factors will also be included in the discussion of κ. In the last part of this chapter, we will further use the calculated effective refractive index to estimate the coupling coefficients κ and the cavity length L. Figure 4.1: The simulated cross section in the DFB laser. 35

50 Figure 4.2: The structure and the refractive index in the simulated III-V/Si hybrid waveguide. W is the thickness of the n-cladding layer. X is the thickness of SCH layer. Y is the thickness of the active region (MQWs). Z is the thickness of the BCB layer. 4.1 Mode Selection in the Passive Waveguide Before we start to discuss the influence of different epilayers on the device performance, we should explain how we choose the optical model in the optical mode calculation. In Fimmwave, the structure we set up is a passive waveguide, which means that we only calculate how the electromagnetic field distributes in the structure with different dielectric materials. Other optical behaviors (e.g., loss, gain, scattering, etc.) are not included. Therefore, you can obtain all the possible optical modes existing in the system. However, in a single mode DFB laser, only the lasing mode can exist in the end [13], and the rest of the modes are useless. It is important to know how to select the mode we need in the calculations. Figure 4.3 (a) is the example structure, Figure 4.3 (b) is the fundamental TE mode, Figure 4.3 (c) is the higher order of TE mode, and Figure 4.3 (d) is the fundamental TM mode. Normally, Fig. 4.3 (b) is the fundamental model in DFB lasers. This mode you see is a supermode. The supermode in the hybrid waveguide can be a linear combination of two fundamental modes existing in the III/V and Si waveguide. The related mathematical method can be found in Ref. [46]. On the other hand, the optical field of high order TE mode does not concentrate in the middle of the hybrid waveguide. There are some fields 36

51 Figure 4.3: (a) The example structure. (b) The fundamental TE mode. (c) The high order TE mode. (d) The fundamental TM mode. distributing near the edge of the hybrid waveguide. However, in real devices, this will make the mode suffer from the large scatter loss due to the roughness of the waveguide sidewall of real laser devices. Consequently, this will increase the threshold of the mode. The mode then can not dominate the lasing mode. Figure 4.3 (d) shows the TM fundamental mode, where the calculated confinement factor is higher than other mode. However, it does not help anything. As I introduced in Chapter 3, the QWs we used here are under the compressive strain. The TE polarization mainly contribute to the light emission under compressive strain. Therefore, the light generation with TM polarization will not dominate the lasing mode. Besides, we found that the device will have two supermodes which are similar, and both of these are the possible candidates of the lasing mode. They are the even and odd mode with different group velocity and effective refractive index. Figure 4.4 shows the vector field of both modes in a 2D plane of the device and the amplitude in the third direction. As the figures show, the even mode has the same direction of the vector fields in the active region and Si region, but the odd mode has the opposite direction of the vector fields in the two waveguides. According to Ref. [13], they have present the mathematical model of the supermodes in a hybrid passive waveguide. The supermode can be expressed as a linear combination of two modes of two uncoupled and separate waveguide with a single 37

52 Figure 4.4: (a) The 2D even TE mode distribution. (b) The odd TE mode distribution. (c) The 2D even TE mode with the amplitude distribution in z-direction. (d) The 2D odd TE mode with the amplitude distribution in z-direction. Figure 4.5: (a) The relation between the mode distribution and the mismatch parameter δ (b) The odd TE mode distribution [13]. 38

53 group velocity: E(x, y, z) =[au 1 (x, y)+bu 2 (x, y)]e iβz, (4.1.1) where a and b are the real number, and the ratio of a can decide the composition of the b supermode. u 1 (x, y) andu 2 (x, y) arethefieldprofileofthetransverse(x,y)direction. Here, u 1 and u 2 are corresponding to the transverse mode profile in the III-V waveguide and the Si waveguide. β is the group velocity of the supermode. Furthermore, they also found that there are even and odd mode coexisting in the hybrid passive waveguide. The expressions are as follows: E o (x, y, z) = iζ [ δ + S u 1(x, y)+u 2 (x, y)]e i( β S)z, (4.1.2) E e (x, y, z) = iζ [ δ S u 1(x, y)+u 2 (x, y)]e i( β+s)z, (4.1.3) 2 β = β 1 + β 2, (4.1.4) 2δ = β 2 β 1, (4.1.5) S = δ 2 + ζ 2, (4.1.6) (4.1.7) where E o and E e are the odd and even mode. β 1 and β 2 are two different group velocities in the two separate (uncouple) waveguide. β 1 and β 2 are corresponding to the group velocities of the III-V waveguide and the Si waveguide. ζ is estimated by the overlap integral of u 1 and u 2,andtheindexperturbationfunction,wherethedetailedcalculation can be found in Ref. [46]. δ is the mismatch parameter. From the Eq , you can clearly see that the group velocity of the even and odd mode are different. According to their calculations [13], the corresponding modes are: 1. δ<0(β 2 <β 1 ): 2. δ>0(β 2 >β 1 ): even : a>b, (4.1.8) odd : b < a, (4.1.9) (4.1.10) even : a<b, (4.1.11) odd : b > a, (4.1.12) where their results show that the even mode dominates the III-V waveguide, and the δ value of the section is smaller than 0 as shown in Fig The even mode dominates the Si waveguide, where the δ value of this section is larger than 0. On the contrary, the odd mode has a totally reverse trend. In order to achieve the single mode operation and couple out the signal easier, researchers use some methods to supress the odd mode and maintain the even mode in the system. One method is to add the absorption region above the taper Si waveguide. Since the odd mode will dominate with the mismatch parameter δ>0area,itwillbeabsorbedlargelyintheabsorptionregionwhichisabovethetaper Si region as shown in Fig. 4.5 (b). Therefore, the threshold of the odd mode will increase. The odd mode can not start lasing, and the even mode dominates the lasing mode. 39

54 Figure 4.6: (a) The normal structure of III-V/Si waveguide with 9QWs (154 nm), 50 nm SCH, 100 nm n-cladding layer and 10 nm BCB. (b) The Y-shaped structure of III-V/Si waveguide. All the epilayers are the same as the normal structure but with different InP structure. (c) The fundamental TE mode of the normal structure and its confinement factor in the active region Γ act. and Si Γ Si. (d) The fundamental TE mode of the Y-shaped structure and its confinement factor in the active region Γ act. and Si Γ Si. 4.2 Different Structures of III-V/Si Hybrid waveguide For achieving a high optical confinement factor in the active region for devices, we have designed a U-shaped structure. This kind of structure increases the upper part of the waveguide, and might increase the optical confinement factors in the active region. The epilayers are shown in Fig. 4.6 (a), the example of the normal structure is comprised of 9 QWs (154 nm), 50 nm SCH, 100 nm n-cladding layer, and 10 nm BCB. All the other epilayers are the same as Fig On the other hand, the Y-shaped structure has the same epilayers as the normal structure but different p-cladding layer. The p-cladding 40

55 Table 4.1: Thickness of different QW numbers QW number thickness (nm) layer in the Y-shaped structure is composed of 5000 nm width of the p-cladding layer with etched 500 nm thickness of the central 2000 nm region as shown in Fig. 4.6 (b). As Fig. 4.6 (c) and (d) show, the Y-shaped structure has the higher optical field in the active region. The calculated confinement factors in the active region Γ act. and Si waveguide Γ Si are % and % in the Y-shaped case. Γ act. and Γ Si in the normal case are % and %. Obviously, the Y-shaped structure has a larger Γ act. which is beneficial for enhancing the operation speed in the laser. Besides, the confinement factors of Y- shaped case have a more average value in both III/V and Si waveguide. However, the Y-shaped structure is hard to be fabricated in the laboratory. As a consequence, we are forced to abandon this good structure. If the technology of the fabrication is improved, the Y-shaped structure is still a potential structure of the applications of optical devices. The structure in the successive discussion will be the normal one. 4.3 Optical Confinement Factor with Different QW numbers In the following session, we will change the thickness of epilayers and see its influence on the optical confinement factors. We will first discuss the different QW number cases, and change the thickness of the layers such as SCH, BCB, and n-cladding layer. Then, some comparisons will be drawn. In the simulation, we set the thickness of the active layers as one material with one refractive index. The thickness of QW and QB are 6 nm and 10 nm, the same as in Chapter 3. The thicknesses for 6, 9, 12 QW cases are 106 nm, 154 nm, 202 nm, and are listed as follows: QWs Figure 4.7 shows the results of 6 QW cases with different epilayers. Figure 4.7 (a), 4.7 (b), 4.7 (c) are the calculated confinement factors of the cases with 100, 150, and 200 nm n-cladding layer. The solid lines are the confinement factors Γ act. in the active region, and the solid lines with hollow circles are the confinement factors Γ Si in the Si region. SCH thickness of the cases are changed from 50, 75, and 100 nm. Furthermore, we also vary 41

56 Figure 4.7: (a) The confinement factors with different SCH thickness and BCB thickness with 100 nm n-cladding layer. (b) The confinement factors with different SCH thickness and BCB thickness with 150 nm n-cladding layer. (c) The confinement factors with different SCH thickness and BCB thickness with 200 nm n-cladding layer. (d) The confinement factors with 10 nm BCB layer, 75 nm SCH and different thickness of n-cladding layer. the thickness of BCB from 10 nm to 60 nm. If the thickness of BCB is too thick, it might reduce the efficiency to couple out the signal generated by the III-V laser. Figure 4.7 (a), 4.7 (b), and 4.7 (c) show that the optical confinement factors Γ act. in the active region are around 6 % to 10.5 %, and Γ Si in the Si region are around 60 % to 72 %. Obviously, there is a large deviation between the two regions, and the number of QW should be increased. Otherwise, the optical field in the active region is too low and will result in asmalldifferentialgainoperation. Roughly,increasingthethicknessoftheSCHlayer can increase the optical confinement factors, and decrease the confinement factors in the Si waveguide at the same time. However, since the thickness of the 6 QWs is too thin, most of the optical fields still remain in the Si waveguide. It does not help too much for increasing the optical fields in the active region to increase the SCH thickness. Increasing the separation between the III-V waveguide and the Si waveguide, which is by increasing the thickness of the bonding layer, will slightly increase the confinement factors in the 42

57 Figure 4.8: (a) The confinement factors with different SCH thickness and BCB thickness with 100 nm n-cladding layer. (b) The confinement factors with different SCH thickness and BCB thickness with 150 nm n-cladding layer. (c) The confinement factors with different SCH thickness and BCB thickness with 200 nm n-cladding layer. (d) The confinement factors with 10 nm BCB layer, 75 nm SCH and different thickness of n-cladding layer. active region in some cases. However, it does not help too much in the 6 QW cases. We also extract the cases with 75 nm SCH, 10 nm BCB with different thickness of n-cladding layers, which are shown in Fig. 4.7 (d). The results do not show a significant difference. It is noticed that the numerical solver might result in some numerical error, but that would probably not change our conclusion. In a short summary, 6 QWs are not enough in the laser system. Even though we try do some modification, the impact is limited QWs The results for 9 QWs with different epilayers are presented in Fig Figure 4.8 (a), 4.8 (b), 4.8 (c) are the calculated confinement factors for the cases with 100, 150, and 200 nm n-cladding layer. The solid lines are the confinement factors Γ act. in the active region, 43

58 and the solid lines with hollow circles are the confinement factors Γ Si in the Si region. The SCH thickness is varied with 75, 100, and 125 nm. Besides, we also vary the thickness of BCB from 10 nm to 60 nm, which still allow to couple the laser signal to the taper Si waveguide. Figure 4.8 (a), 4.8 (b), and 4.8 (c) show that the optical confinement factors Γ act. in the active region are from 17 % to 35 %, and Γ Si in the Si region are around 15 % to 50 %. In most of the calculated cases, increasing the thickness of the SCH thickness will increase the confinement factors in the active region, and decrease the confinement factors in the Si region. Since the active region is relative thin, the commercial and laboratory products usually utilize the SCH layer to enhance the optical confinement factor in the active region. The material of a SCH layer usually has a lower refractive index than the active region but a quit high refractive index. The SCH layers work as we expected in our simulation results. Increasing the SCH thickness increase Γ act. and decrease Γ Si. From these figures, the optical confinement factors in the active region increase with increasing the thickness of the bonding layer. The separation decomposes the supermodes, and more fields tend to concentrate in the III/V waveguide. Figure 4.8 (d) shows the cases of 75 nm SCH and 10 nm BCB with different thickness of the n-cladding layers. The calculated results show that increasing the thickness of n-cladding layer slightly decreases the confinement factors in both the active region and Si waveguide. With enlarging the distance between the active region and the Si region, the field will tend to concentrate in the III/V waveguide due to its high refractive index. Then, the confinement factors in the Si waveguide decrease. However, the increasing thickness of the n-cladding layer might share some of the fields in the III/V waveguide, and the confinement factor in the active region therefore slightly decreases with increasing the n-cladding layer QWs We further increase the QW number to 12. Figure 4.9 (a), 4.9 (b), 4.9 (c) are the calculated confinement factors for the cases with 100, 150, and 200 nm n-cladding layer. The solid lines are the confinement factors Γ act. in the active region, and the solid lines with hollow circles are the confinement factors Γ Si in the Si region. Each of the cases is comprised of a different SCH thickness, which are 25, 50, 75 nm. Furthermore, we also vary the thickness of BCB from 10 nm to 60 nm. Figure 4.9 (a), 4.9 (b), and 4.9 (c) show that the optical confinement factors Γ act. in the active region are from 30 % to 50 %, and Γ Si in the Si region are around 10 % to 50 %. Because the thickness of the active region is really thick, we use thinner SCH layers than the 9QW and 6QW cases in these calculations. The results show that the elevated thickness of the SCH layer enhances the optical confinement factors in the active region. The reason is the same as for the case of 9 QWs. On the other hand, increasing the bonding layer thickness enhances the optical confinement factors in the active region as well. As we separate the hybrid waveguide more, the optical field tends to concentrate in the active region which has higher refractive index. The fields of the supermode, which is dominated by the fundamental mode in the Si waveguide, thus decreases. Figure 4.9 (d) shows the results for the cases with 75 nm SCH, 10 nm BCB, and different thickness of n-cladding layers. The results indicate that the confinement factors in both the active and Si region decrease with increasing thickness of the n-cladding layer. The decreasing confinement factors in the Si region are attributed to the increasing volume of the III/V waveguide. However, the enlarged part of the n- cladding layer shares some of the optical fields with the active region, and results in a 44

59 Figure 4.9: (a) The confinement factors with different SCH thickness and BCB thickness with 100 nm n-cladding layer. (b) The confinement factors with different SCH thickness and BCB thickness with 150 nm n-cladding layer. (c) The confinement factors with different SCH thickness and BCB thickness with 200 nm n-cladding layer. (d) The confinement factors with 10 nm BCB layer, 75 nm SCH and different thickness of n-cladding layer. smaller confinement factors in the active region Comparison of Confinement Factors with Different QW Numbers To know how the thickness of the active region influences the calculated confinement factor Γ act., we chose the cases with 75 nm SCH, 10 nm BCB, 100 nm n-cladding layer, and different QW numbers. Figure 4.10 shows that the confinement factors Γ act. in the active region are 9.09 %, %, and %, which are corresponding to the 6 QW, 9 QW, and 12 QW cases respectively, and the values of Γ Si in the Si region are %, %, and %, which are corresponding to the 6 QW, 9 QW, and 12 QW 45

60 Figure 4.10: (a) The selected structures with 75 nm SCH, 10 nm BCB, 100 nm n-cladding layer, and different QW numbers. (b) The confinement factors of different QW number cases. (c) The calculated confinement factor in a single QW with different QW numbers. cases respectively. Obviously, increasing the volume of the active regions elevates the confinement factor Γ act..wealsocalculatedtheconfinementfactorpereveryqwinthe different QW number cases. Figure 4.10 (c) shows that the optical confinement factors in single QW of each case are %, %, %. The 12 QW case shows the highest confinement factors per QW. 4.4 Coupling coefficient The product of the coupling coefficient and the cavity length κl in Eq is regarded as the degree of the reflection in DFB lasers as I introduced in the section DF B Lasers of Chapter 1. In practical designs, we keep the value of κl from 1 to 3 corresponding to 58% to 99% of the reflectivity R. Here, we used the Eq to estimate the κ values of our cases. The grating order m is 1, and the duty cycle l h Λ is 50 %. Figure 4.11 shows the different simulated grating cross section. The height of the higher effective refractive 46

61 Figure 4.11: The simulated cross section in the discussion of the coupling coefficient κ discussion. index n h eff region ( with nh eff the effective refractive index in the Si rib part) is 400 nm. The height of the lower effective refractive index n l eff region, which is the trench section, is 220 nm. In the following paragraph, we will show and analyze the calculated results of 6, 9, and 12 QW cases. In order to make readers know more about the influence of the parameter adjustments, some comparisons will be shown. In the end, the relation of the cavity length L and κ will be discussed QWs Figure 4.12 (a), 4.12 (c), and 4.12 (e) are the calculated n h eff and nl eff of the fundamental mode with 400 and 220 nm height Si rib (b), 4.12 (d), and 4.12 (f) are the calculated κ values and the confinement factors Γ act..wefirsttakeaglimpseonthecalculated refractive index n h eff and nl eff.theresultsofnl eff with 220 nm Si shows that increasing the SCH thickness will increase the effective refractive index n l eff. Since the refractive index in the active region is higher than the Si region 3.54, increasing SCH thickness will make more fields concentrate in the active region and then elevate the effective refractive index n l eff.besides,increasingthebcbthicknessdoesnotchangetheeffective refractive index n l eff significantly, because most of the fields are already in the active region. Normally, the effective refractive index n h eff will be higher than nl eff,becausethe optical fields are more in the active region, where the refractive index of the active region is higher than the Si region. Our calculated optical confinement factors Γ act. in different cases with different SCH thickness are too close. The effective refractive index n h eff does not have a significant relation with the various SCH thickness. The calculated refractive index n h eff is lower with increasing the BCB thickness. On the other hand, the coupling coefficient κ increases with decreasing SCH thickness due to the effective refractive index 47

62 Figure 4.12: (a), (c), and (e) are the calculated nhef f and nlef f of the fundamental mode with 400 and 220 nm height of the Si rib. (b), (d), and (f) are the calculated κ values and the confinement factors Γact.. nlef f, and a decreasing BCB thickness also increases κ. This means that more optical fields in the Si region will enhance the reflection in the grating. Thus, it will have the same degree of reflection but with shorter cavity length under the high κ value. On the other side, we also draw a comparison of the κ values with different n-cladding layers. Figure 4.13 (a) shows the structures, which are composed of 75 nm SCH, 10 nm BCB, 106 nm active region (6 QWs), and different thickness of n-cladding layers. Their coupling coefficients κ are presented in Fig (b). The result shows that the coupling coefficient will decrease with increasing the thickness of the n-cladding layer. The values are 1263, 1115, and 1007 cm 1 in the cases with 100 nm, 150 nm, and 200 nm n-cladding 48

63 Figure 4.13: (a) The thickness of the epilayers with different n-cladding layers W nm. The cases in the comparison are composed of 75 nm SCH, 10 nm BCB, 106 nm active region (6 QWs), and with different thickness W nm of n-cladding layers. (b) The coupling coefficients κ with different n-cladding layers. layers. From the definition of the reflectivity in Eq , a higher κ results in a shorter cavity length L under the same reflectivity. According to the definition f r in Eq , the shorter cavity length L will increase the speed of the laser. Therefore, a thinner n-cladding layer will be desired QWs The calculated n h eff and nl eff of the fundamental mode with 400 and 220 nm height Si rib are shown in Fig (a), 4.14 (c), and 4.14 (e). Figure 4.14 (b), 4.14 (d), and 4.14 (f) are the calculated κ values and the confinement factors Γ act..theresultsofn l eff with 220 nm Si shows that increasing the SCH thickness will increase the effective refractive index n l eff due to the higher confinement in the active region. Because most of the fields are in the active region, there is no significant change of n l eff with various thickness of the BCB layers. On the other hand, n h eff will be higher with increasing the SCH thickness, when the BCB thickness is larger than 30 nm. Below 30 nm BCB thickness, the trend is not significant. Also, n h eff increases with decreasing the BCB thickness. Consequently, the coupling coefficient κ increases with decreasing SCH and BCB thickness. From Fig. 4.8, increasing the optical confinement factors Γ act. usually decreases the confinement factors Γ Si at the same time. Thus, the coupling coefficients κ have a reverse trend with the trend of Γ act. ;namely,thedegreeofthereflectionofthegratingishighlyrelatedtopenetration of the optical fields in the 400 nm Si region. Furthermore, the comparison of the κ values with different n-cladding layers is shown in Fig (b), and their structures are shown in Fig (a). We found a significant drop of the κ values with increasing the n-cladding layers. The values are 849.4, 674.1, and cm 1 in the cases with 100 nm, 150 nm, and 200 nm n-cladding layers respectively. The κ value of the case with 100 nm n-cladding layer is 1.56 times higher than the case 49

64 Figure 4.14: (a), (c), and (e) are the calculated nhef f and nlef f of the fundamental mode with 400 and 220 nm height Si rib. (b), (d), and (f) are the calculated κ values and the confinement factors Γact.. with 200 nm n-cladding layer. Consequently, we suggest that it is better to choose the thinner n-cladding layer in the 9 QW case for the high speed laser QWs Figure 4.16 (a), 4.16 (b), and 4.16 (c) show the calculated nhef f and nlef f. Figure 4.16 (b), 4.16 (d), and 4.16 (f) are the calculated κ values and the confinement factors Γact.. In the 50

65 Figure 4.15: (a) The thickness of the epilayers with different n-cladding layers W nm. The cases in the comparison are composed of 75 nm SCH, 10 nm BCB, 154 nm active region (9 QWs), and with different thickness W nm of n-cladding layers. (b) The coupling coefficients κ with different n-cladding layers. effective refractive index calculations, increasing the SCH thickness increase both of the n h eff and nl eff. But, in the nl eff calculations, changing the BCB thickness does not vary n l eff much, because most of the optical fields are already in the active region. The results of κ show that increasing the SCH thickness and increasing the BCB thickness will result in a lower κ value, where Γ act. has a reverse trend. The reason is the same as I mentioned before. Decreasing the optical confinement factor Γ Si in the Si region will increase the Γ act. in the active region and decrease the coupling coefficient. Acomparisonofκ with 75 nm SCH, 10 nm BCB, 12 QWs, and different n-cladding are shown in Fig (b). From the calculations, we found that increasing the thickness of the n-cladding layer will result in a lower κ value. The values are 520, 364.9, and cm 1 in the cases with 100 nm, 150 nm, and 200 nm n-cladding layers respectively. The κ value of the case with 100 n-cladding layer is almost 2 times higher than the case with 200 nm n-cladding layer. Besides, in Fig. 4.9 (d), the optical confinement factor in the active region for the case with 100 nm n-cladding layer is significantly higher than the case with 200 nm n-cladding layer. Since the κ and Γ act. for the case with 100 n-cladding layer are higher, we suggest that we should choose a thinner n-cladding layer Comparison of Coupling Coefficient with Different QW Numbers In this subsection, the structures we chose in Fig (a) are 75 nm SCH, 10 nm BCB, 100 nm n-cladding layer, and the different numbers of QW. The difference of the κ values between different cases are presented in Fig (b). The relation of the cavity length L and κl in different cases are shown in Fig (c). In Fig (b), we can clearly see 51

66 Figure 4.16: (a), (c), and (e) are the calculated n h eff and nl eff of the fundamental mode with 400 and 220 nm height Si rib. (b), (d), and (f) are the calculated κ values and the confinement factors Γ act.. that the increasing number of the QWs drastically decreases the κ value. The value of the 12 QW case is 5 times smaller than the 6 QW case. In Fig. 4.18, we can see that the calculated cavity lengths L with the κl value 2 of 6, 9, and 12 QWs are µ m, µ m, and µ m, respectively. Although increasing the QW number has a higher optical confinement factor it also elevates the cavity length due to the variation of the coupling coefficient. The lower cavity length of the low QW number cases might be beneficial for enhancing the speed of the laser and minimizing the device size. But, the ultra short cavity length implies that the thermal resistance will be higher in the device and thus 52

67 Figure 4.17: (a) The thickness of the epilayers with different n-cladding layers W nm. The cases in the comparison are composed of 75 nm SCH, 10 nm BCB, 202 nm active region (12 QWs), and with different thickness W nm of n-cladding layers. (b) The coupling coefficients κ with different n-cladding layers. worsen the device performance. According the thesis written by Dr. Stevan Stanković in 2012 [44], he suggested that κ lower than 150 cm 1 is suitable for the design. However, our calculated results are far higher than this value. Furthermore, from Ref. [47], they estimate the threshold gain g grating by solving the eigenvalue equations of the propagation of the backward and forward electromagnetic waves. g grating should be expressed as: g grating = 2π2 κ 2 L 3. (4.4.1) If κl =2,thenthecavitylengthL can be expressed as Eq As a result, our calculated cavity lengths will result in a very large threshold gain in the numerical calculations, which is unreasonable for the reality. For example, g grating will be around 1230 cm 1 in 6QWcases,whichisextremelylargeforthedevice. Therefore,wewillusethecavity length evaluated by Eq with κl =2inthefollowinganalysisrelatedtothecavity length. However, Fig (c) still makes us to take a glimpse of the influence of QW numbers to the cavity length. L = 2π2. (4.4.2) 2 2 g grating 53

68 Figure 4.18: (a) The structures are composed of 75 nm SCH, 10 nm BCB, 100 nm n-cladding layer, and the different numbers of QWs. (b) The coupling coefficients κ vs. different QW numbers. (c) The coupling coefficients κl vs. the cavity length with different QW cases. 54

69 Chapter 5 Optical Gain and Differential Gain Calculation The optical gain and the differential gain are very important physical indicators for estimating the performance of high speed lasers. We will compare the calculations with different injected carrier density in different QW number cases, so that we can know the influence of the QW number to the optical gain. Then, the differential gain which is related to the speed of the lasers can be estimated by the optical gain data. As I stated in Chapter 1, the oscillation frequency determining the speed of lasers is expressed in Eq. dg/dn The value is proportional to,whereboththedifferentialgainandthe N wd ww actl dg/dn N wd ww actl volume of the total QW layers should be considered. A comparison of for the different QW number cases will be drawn by considering both the simulated differential gain and the total QW volume. In the end, we will show the calculations of the differential gain and volume of all the structures calculated in Chapter 4 to see how these calculated parameters will influence the speed. Our input parameters are shown in Fig The interpolation method of the material parameters can be found in Ref. [38]. The structures we selected are shown in Fig. 5.1 (a). The structures are composed of 75 nm SCH, 10 nm BCB, 100 nm n-cladding layer, and Table 5.1: Detailed material parameters binary material GaAs AlAs InAs GaP InP m e /m γ γ γ ϵ r E p (ev) m e /m 0 is the relative effective electron mass to bare electron mass. γ 1,2,3 are Luttinger parameters which are utilized to estimate the effective mass of holes. ϵ r is the relative dielectric constant of the materials. E p is the energy parameter that depends on the material. 55

70 Figure 5.1: (a) The structures are composed of 75 nm SCH, 10 nm BCB, 100 nm n- cladding layer, and the different numbers of QWs. (b) The optical confinement factors of the total QW layers in the cases with 6, 9, and 12 QWs. the cases with different numbers of QWs. Figure 5.1 (b) shows the optical confinement factor of the total QW layers in the cases with 6, 9, and 12 QWs, which are %, %, and % respectively. The 12 QW case has the highest total optical confinement factor in the QW layers. Figure 5.2 (a) is the scheme of the simulated structures. Figure 5.2 (b), 5.2 (c), and 5.2 (d) are the calculated optical gain curves with different injected carrier density in 6, 9, 12 QW cases respectively. The injected carrier densities (equal amount of electrons and holes) in all the cases range from to cm 3 with an interval of cm 3. As the figures show, we can see that the peak gain wavelength in all the curves under the low injected carrier density ( cm 3 )is1575nm. InChapter3,the calculated energy gap between the ground states of electrons and holes is ev which is corresponding to 1575 nm wavelength. When we increase the injected carrier density, the low level states will be filled, and the carriers will tend to fill the higher level states. Consequently, the peak positions have a blue shift in all the cases, while we inject more carriers into the QWs. At higher carrier densities we observe that the increase of the gain with carrier density becomes slower. This phenomenon implies that the differential gain decreases with increasing gain. Besides, you can easily see that the gain values at the same injected carrier density are higher for a larger number of QWs. We further extract the gain value at 1550 nm wavelength in Fig The result clearly shows that the case with the higher QW number has a higher gain value due to its higher optical confinement factor in the total QW layers. For a more detailed point of view, we have estimated a threshold gain g th. As we know, the threshold gain is the gain value that the laser start to lase. Therefore, we assume that the total loss is 20 cm 1 and the threshold induced by the distributed back grating is 20 cm 1. Therefore, g th should be 40 cm 1 by our assumption. The threshold carrier densities, which are the corresponding carrier density of g th,are cm 3, cm 3,and cm 3,where 56

71 Figure 5.2: (a) The structures are composed of 75 nm SCH, 10 nm BCB, 100 nm n- cladding layer, and different QW numbers. (b) The optical gain curves of the 6 QW case. (c) The optical gain curves of the 9 QW case. (d) The optical gain curves of the 12 QW case. The injected carrier density of the equal amount of electrons and holes in all the cases are from to All the interval of injected carrier density between the two adjacent gain curves in all the figures are they are corresponding to the case with 6 QWs, 9 QWs, and 12 QWs. The threshold carrier density of the case with a higher QW number is smaller due to the higher optical confinement factor in the total QW layers. Figure 5.4 shows the calculated differential gain at 1550 nm wavelength. The corresponding different gain values at g th of the 6 QWs, 9 QWs, and 12 QWs are cm 2, cm 2,and cm 2.Wefoundthatthedifferentialgainofthe12QW case are 21.7 times larger than the 6 QW case and 2.89 times larger than the 9 QW case. Besides, the differential gain under the higher operated gain, which also means the higher operated carrier density, is lower. Therefore, the operated carrier density should not be too high over the threshold gain. Otherwise, the speed of the lasers will be low. Figure 5.5 shows the estimated dg/dn N wd ww actl values of different QW cases at 1550 nm wave- 57