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Stress engineering with silicon nitride stressors for Ge-on-Si lasers Ke, Jiaxin 2017

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  STRESS ENGINEERING WITH SILICON NITRIDE  STRESSORS FOR GE-ON-SI LASERS by  Jiaxin Ke  B.A., The University of Science and Technology of China, 2015  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Material Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)   June 2017  ยฉ Jiaxin Ke, 2017 ii Abstract Silicon compatible lasers are in great need for applications such as on-chip and short-reach optical interconnects. Although InAs/GaAs quantum dot lasers monolithically grown on Si have been realized and are well-performed, due to material contamination issues, it is time and cost intensive for those III-V materials to enter mainstream Si processing facilities. Germanium(Ge)-on-Silicon(Si) laser is promising as a solution to solve the Si-compatible laser problem as it is compatible with Si processing. So far, the main problems in Ge lasers are that they have a high threshold current density and low efficiency. Laser structure designs with top and side silicon nitride stressors were proposed in this work and shown to be effective in reducing the threshold current (Ith) and improving the wall-plug efficiency (ฮทwp) of Ge-on-Si lasers. Side stressors turned out to be a more efficient way to increase ฮทwp than using the top and side stressors together. With the side stressors and geometry optimizations, a maximum ฮทwp of 34.8% and an Ith of 36 mA (Jth of 27 kA/cm2) were achieved with a defect limited carrier lifetime (๐œ๐‘,๐‘›) of 1 ns. With ๐œ๐‘,๐‘› being 10 ๐‘›๐‘  , an Ith of 4 mA (Jth of 3 kA/cm2) and a ฮทwp of 43.8% were achieved.  These are tremendous improvements from cases without any stressors. Compared to other stress introduction methods, such design is much more suitable for Ge laser structure implementation. These results provide a strong support to the Ge-on-Si laser technology and create an effective way to improve the Ge laser performance.     iii  Lay Summary Laser is a device that can emit a single color of light and is suitable for communication applications. Germanium (Ge)-on-Silicon (Si) laser is a new type of laser that can be made with low cost and is promising in making our computer run faster and communicate faster with each other. However, the performance of Ge-on-Si lasers has so far been insufficient. To make it perform better, we designed a new structure and optimized it with stress engineering method to investigate the performance potential of Ge-on-Si lasers. Our computational simulations indicate that method is promising to improve Ge lasersโ€™ energy efficiency and reduce the current required for lasing.  iv Preface The work in this thesis was funded by UBC. All of the work was done in the Department of Materials Engineering at UBC during the past two years, including the structure designs, stress simulation and the laser modeling and optimizations. All the theoretical modeling and simulations were conducted by the author. In addition, one journal article and one conference paper, and one conference abstract have been published based on the work in Chapter 4 to 5: Ke, Jiaxin, Lukas Chrostowski, and Guangrui Xia. "Stress engineering with silicon nitride stressors for Ge-on-Si lasers." IEEE Photonics Journal 9.2 (2017): 1-15. J. Ke, G. M. Xia, and L. Chrostowski, "Structure and stress engineering for Ge-on-Si lasers using silicon nitride stressors," 2016 IEEE 13th International Conference on Group IV Photonics (GFP), Shanghai, 2016, pp. 28-29. J. Ke, Z. Li, S. Li, and G. Xia, "Stress and structure optimizations with side silicon nitride stressors for Ge-on-Si lasers," in Advanced Photonics 2016 (IPR, NOMA, Sensors, Networks, SPPCom, SOF), OSA technical Digest (online) (Optical Society of America, 2016), paper IW1A.5.  A different topic of stress solver developing was done in Lumerical Solutions, Inc., which was not discussed in this thesis.     v  Table of Contents Abstract ........................................................................................................................................... ii Lay Summary ................................................................................................................................. iii Preface............................................................................................................................................ iv Table of Contents ............................................................................................................................ v List of Tables ............................................................................................................................... viii List of Figures ................................................................................................................................ ix List of Symbols ............................................................................................................................ xiii List of Abbreviations ................................................................................................................. xviii Acknowledgements ...................................................................................................................... xix Dedication ..................................................................................................................................... xx Chapter 1: Introduction ................................................................................................................... 1 1.1 Mooreโ€™s law and the interconnect bottleneck ............................................................... 1 1.2 Optical interconnects, Si photonics, and Si-compatible lasers ..................................... 2 1.3 Types of Si-compatible lasers ....................................................................................... 4 1.4 Structure of this thesis ................................................................................................... 6 Chapter 2: Literature Review and Problem Definition ................................................................... 8 2.1 Band gap engineering methods ..................................................................................... 8 2.2 Doping technologies ................................................................................................... 11 2.3 Stress engineering techniques for Ge .......................................................................... 13 vi 2.4 Progress of Ge lasers ................................................................................................... 16 2.5 Thesis Objective.......................................................................................................... 17 Chapter 3: General Laser Theories ............................................................................................... 19 3.1 Introduction of Fabry-Perot lasers and rate equation .................................................. 19 3.2 Threshold condition .................................................................................................... 22 3.3 Optical loss mechanisms ............................................................................................. 23 3.4 Loss mechanism for carriers ....................................................................................... 25 3.4.1 Spontaneous radiative recombination .............................................................. 25 3.4.2 SRH recombination ......................................................................................... 26 3.4.3 Auger recombination ....................................................................................... 27 3.5 Gain ............................................................................................................................. 30 Chapter 4: Ge Laser Related Modeling and Parameters ............................................................... 33 4.1 Doping-induced Ge band gap changes ....................................................................... 33 4.2 Stress effect and models.............................................................................................. 35 4.3 Effective mass of ฮ“ conduction band .......................................................................... 38 4.4 Figures of merits of lasers ........................................................................................... 41 4.4.1 Threshold current ............................................................................................. 41 4.4.2 Slope efficiency ............................................................................................... 41 4.4.3 Maximum wall-plug efficiency ....................................................................... 42 vii  4.5 Modeling with LASTIPTM and model calibrations ..................................................... 43 4.6 Modeling of Ge laser with stressors ............................................................................ 47 4.7 Limitations of our modeling ....................................................................................... 55 Chapter 5: Structure and Strain Optimizations ............................................................................. 58 5.1 Optimizations of structure 1 without stressors ............................................................ 59 5.2 Optimizations of structure 2 with side stressors only ................................................. 61 5.2.1 Polysilicon thickness (dpoly) optimization ........................................................ 61 5.2.2 Ge Width (W) optimizations ........................................................................... 63 5.2.3 Ge thickness dGe optimization ......................................................................... 66 5.2.4 Strainโ€™s Impact ................................................................................................. 69 5.3 Optimizations of structure 3 with top and side stressors ............................................ 73 5.4 Comparisons of the structures ..................................................................................... 75 5.5 Effect of defect-limited minority carrier lifetime on the performance ....................... 78 Chapter 6: Thesis Summary and Suggestions for Future Work ................................................... 81 Bibliography ................................................................................................................................. 83  viii List of Tables Table 3-1 Threshold energy for different Auger recombinations. Table courtesy of Rodolfo E. Camacho-Aguilera at MIT [30]. ................................................................................................... 30 Table 4-1 Material parameters used in the modeling of Ge laser [26] ......................................... 46 Table 4-2 Laser performances of the three structures in Figure 4-11. .......................................... 55 Table 5-1 Laser performance of the 3 structures in Figure 5-16 after structure optimizations. ... 76 Table 5-2    Laser performance of the three structures in Figure 5-20. ........................................ 80    ix  List of Figures Figure 1-1 RC delay time of metal wire and intrinsic gate delay vs. technology node [4]. ........... 2 Figure 1-2 Key devices in silicon photonics [9]. ............................................................................ 4 Figure 2-1 Band structures of Ge under different conditions. (a) bulk Ge without strain or doping; (b) tensile strained Ge without doping; (c) highly n-type doped Ge without strain. Figure courtesy of Donguk Nam at Stanford University [29]. ................................................................................. 9 Figure 2-2 Calculation and experimental result of band crossing of GeSn alloy as a function of Sn concentration. Figure courtesy of Huo Yijie at Stanford University [3]. ..................................... 11 Figure 2-3 Top SiN on Ge stripes for strain introduction  [41]. Figure reproduced with the permission from Optical Society of America. .............................................................................. 15 Figure 2-4 Suspended Ge microbridges with high tensile strain  [44]. Figure reproduced with the permission from OSA Photonics Research. .................................................................................. 16 Figure 2-5 Prototype of the electrically pumped Ge-on-Si laser device [23]. Figure reproduced with the permission from Optical Society of America. ................................................................ 17 Figure 3-1 Band diagram of a forward biased double-heterostructure laser diode [46]. .............. 19 Figure 3-2 Diagram of CCCH (a) direct and (b) indirect Auger recombination. Figure courtesy of Rodolfo E. Camacho-Aguilera at MIT [30]. ................................................................................. 28 Figure 3-3 Radiative carrier lifetime of n-type and p-type dopants in Ge. Figure courtesy of Rodolfo E. Camacho-Aguilera at MIT [59] [30]. ......................................................................... 29 Figure 4-1 n-type band gap narrowing effect of different models at 300 K. ................................ 35 Figure 4-2 Direct and indirect band gap energies under different biaxial strain. ......................... 37 Figure 4-3 LASTIPTMโ€™s results of material gain at different mฮ“* with 0.25% biaxial strain carrier concentration n=p=4ร—1019cm-3. .................................................................................................... 39 x Figure 4-4 Ge profiles measured by SIMS. The Ge profiles are shifted laterally for easy comparison. UGUS: undoped-Ge/undoped-Si; UGPS: undoped-Ge/ Phosphorus-doped-Si; PGUS: Phosphorus-doped-Ge/undoped- Si; PGPS: Phosphorus-doped-Ge/Phosphorus-doped-Si. The black dash line is the Ge profile of sample PGPS before annealing. Figure courtesy of Feiyang Cai at UBC [74]. .................................................................................................................................. 40 Figure 4-5 Cross-section of the Ge-on-Si heterojunction laser structure simulated. .................... 47 Figure 4-6  L-I curves for the experimental result, calibration result, and sensitivity tests with a smaller FCA loss: ๐›ผ๐‘– = 5.0 ร— 10โˆ’19N + 0.923 ร— 10โˆ’17P, and a smaller effective mass meฮ“* = 0.0453 me. ..................................................................................................................................... 47 Figure 4-7 Laser structure simulated (cavity width = 1 ฮผm, thickness = 0.2 ฮผm, length = 270 ฮผm, cladding thickness = 0.18 ฮผm). (a) Structure 2: with side nitride stressors; (b) Structure 3: with top & side nitride stressors. The width of the top nitride stressor is the same as that of the Ge cavity. The metal contacts are composed of Ti and Al same as those in Ref. [23], shown in Figure 4-6.48 Figure 4-8 Normalized light intensity of 1D cut in the centre line of (a) structure 2 (structure 1 have a similar distribution); (b) structure 3. 2D profile of (c) structure 1; (d) structure 2; (e) structure 3...................................................................................................................................... 49 Figure 4-9 2D Strain map on the cross section: (a) ฮตeb (b) ฮตzz of structure 2 with side stressors only (cavity width = 1 ฮผm, thickness = 0.2 ฮผm, cladding thickness = 0.18 ฮผm). ................................ 51 Figure 4-10 2D Strain map on the cross section: (a) ฮตeb (b) ฮตzz of structure 3 with side and top stressors (cavity width = 1 ฮผm, thickness = 0.2 ฮผm, cladding thickness = 0.18 ฮผm). .................. 52 Figure 4-11 L-I curve comparison for the three structures before the structure optimizations. ... 54 Figure 5-1 Ith and ฮทwp of Structure 1, dpoly dependence. ................................................................ 59 Figure 5-2 Ith and ฮทwp of Structure 1 (a) width dependence; (b) dGe dependence. ........................ 60 xi  Figure 5-3 Polysilicon thickness dpoly dependence (W =1 ฮผm, dGe = 0.2 ฮผm) of (a) <ฮฑi> and ฮทext ,ฮทd  (b) Ith and ฮทwp. ............................................................................................................................... 62 Figure 5-4 Ge width W dependence (dpoly =0.8 ฮผm, dGe = 0.2 ฮผm) of (a) strain ฮตeb and ฮ“ (b) <ฮฑi> and ฮ“. ............................................................................................................................................ 63 Figure 5-5 Ge width W dependence (dpoly =0.8 ฮผm, dGe = 0.2 ฮผm) of  (a) ฮทd and ฮทext (b) Ith and ฮทwp........................................................................................................................................................ 64 Figure 5-6 Impacts of Ge width (W) on other parameters. ........................................................... 65 Figure 5-7 Ge thickness dGe dependence (W =0.5 ฮผm, dpoly = 0.8 ฮผm) of Ith and ฮทwp. ................. 66 Figure 5-8 Ge thickness dGe dependence (W =0.5 ฮผm, dpoly = 0.8 ฮผm) of (a) Strain ฮตeb and ฮ“ (b) <ฮฑi> and ฮ“. .................................................................................................................................... 67 Figure 5-9 Ge thickness dGe dependence (W =0.5 ฮผm, dpoly = 0.8 ฮผm) of  ฮทd and ฮทext. ................ 68 Figure 5-10 Impacts of Ge thickness dGe on other parameters. .................................................... 68 Figure 5-11 (a) Strain ฮตeb  and ฮป with different dGe (W=0.5ฮผm, dpoly = 0.8ฮผm): (b) direct band alignment under different strain at thermal equilibrium (V=0, dGe=0.8 ฮผm). ............................... 70 Figure 5-12 Strain impact with different dGe (W=0.5ฮผm, dpoly = 0.8ฮผm): (a) material gain at different strain with carrier concentration n=p=4ร—1019cm-3, (b) Ith, and (f) ฮทd. ............................ 71 Figure 5-13 Strain impact with different dGe (W=0.5ฮผm, dpoly = 0.8ฮผm): (a) ฮทext (b) ฮทd. ............. 72 Figure 5-14 Ith and ฮทwp  of Structure 3 (a) width dependence (b) dGe dependence. ...................... 74 Figure 5-15 Ith and ฮทwp  of Structure 3, dpoly dependence. ............................................................. 75 Figure 5-16 L-I curve for three structures after optimization. ...................................................... 76 Figure 5-17 Current density field of structure 2 (a) in x direcdtion; (b) in y direction at 60 mA inject level. .................................................................................................................................... 77 xii Figure 5-18 Current density field of structure 3 (a) in x direcdtion; (b) in y direction at 60 mA inject level. .................................................................................................................................... 77 Figure 5-19 Defect limited carrier lifetime dependence of Ith and ฮทwp. ........................................ 79 Figure 5-20 . L-I curve for three structures with ฯ„๐‘,๐‘› = 10 ns. .................................................... 80  xiii  List of Symbols A Free electron absorption coefficient. aฮ“, aL Hydrostatic deformation potential at ฮ“ and L valleys. aeff Effective hydrostatic deformation potential. B Free hole absorption coefficient. B21, B12 Einstein coefficients for stimulated emission and absorption. b Shear (uniaxial) deformation potential. beff Effective shear (uniaxial) deformation potential. C11, C12 Elastic constants. Cn, Cp Electron and hole Auger recombination coefficient. cnj, cpj Electron and hole capture coefficients of the jth deep trap. d Thickness of laser cavity. dGe Thickness of Ge. dPoly Thickness of polysilicon. Efc, Efv Quasi-Fermi level of electrons in the conduction and valence band. Ec, Ev Energy levels in the conduction and valence band that associated with the photon transition and ๐ธ๐‘ โˆ’ ๐ธ๐‘ฃ = โ„๐œ”. Eg Band gap energy. Eos Stored optical energy in the cavity. Eฮ“-hh๏ผŒEฮ“-lh Direct band gap energies between the ฮ“ valley and the heavy hole, light hole valence band. EL-hh, EL-lh Indirect band gap energies between L valley and the heavy hole, light hole valence band. xiv EC-hh, EC-lh Band gap energies between conduction band and the heavy hole, light hole valence band. โˆ†EgL   Indirect band gap narrowing effect. ฮดEhy, ฮดEsh  Hydrostatic and shear deformation energy. ftj Occupancy possibility of the jth deep trap level. fc, fv Electronโ€™s occupation probability in the conduction band and valence band. Ggen Electrons genertaion rate. g Material gain. gth Threshold optical gain. โ„Ž, โ„  Planck constant and reduced Planck constant. I Injection current. Ith Threshold current. Iop Operation current. Jth Threshold current density. K Wave vector. k Boltzmann constant. L Length of Ge cavity. mn, mp Bulk effective masses for electrons and holes. mr* Reduced effective mass. mc*, mv* Effective mass of direct conduction and valence band. me Free electron mass. N Electron density. Nth Threshold carrier concentration. xv  Ntj Density of the jth deep trap.  N1j Electron concentration when the electron quasi-Fermi level coincides with the energy level. Ni  Intrinsic carrier density without doping. ND  n-type dopantsโ€™ concentration. n Refractive index. P Hole density. P1j Hole concentration when the electron quasi-Fermi level coincides with the energy level Etj of the jth trap.  Pth Threshold hole density of hole. Pout The optical power output. Pelect, Popt Electric and optical power. |pcv| Transition matrix factor. Qฮต  Shear deformation energy.  q Electron charge. Rrec  Electron recomination rate. Rsp Spontaneous recombination rate. RSRH SRH absorption rate. RAug  Auger recombination rate. Rst  Stimulated emission rate. Rec Carrier desity ralated recombination. R1, R2 Reflectivity of two end facets. R21 Stimulated emission rate from energy level 2 to 1. xvi R12 Stimulated absorption of photons from energy level 1 to 2. Rntj, Rptj  Electron and hole recombination rates through jth deep trap level. Etj of the jth trap.  R  Series resistance. S Photon densitiy. T Temperature. Vol, Volp Active region volume for carriers and photons. Vop The operation voltage. VD Diode voltage. vg Group velocity. W Width of Ge cavity. ฮฑi, ฮฑm Internal loss and mirror loss coefficient. <ฮฑi> Weighted average of the internal loss. ฮฒ Ratio of vertical strain and effective biaxial strain which is defined as ฮฒ=ฮตzz/ฮตeb. ฮ” Spin-orbit splitting energy. ฮตxx, ฮตyy, ฮตzz Strain in horizental, length  and vertical direction. ฮตb, ฮตeb Biaxial strain and effective biaxial strain. ฮ“ Confinement factor. ฮทi, ฮทd, ฮทext Internal efficiency, slope efficiency and extraction efficiency. ฮทwp, Wall-plug efficiency. ฮป Light wavelength. ฯƒnj, ฯƒpj Electron and hole capture cross sections of the jth deep trap.   ฯr  Joint density of states of direct conduction band. ฯ„nj, ฯ„pj Electron and hole lifetime due to the jth trap. xvii  ฯ„pho Photon lifetime. ฯ„p,n Defect limited carrier lifetime for hole and electron. ฯ…ฬ…n, ฯ…ฬ…p  Average thermal velocity of electrons and holes. ฯ‰ Angular frequency.                          xviii List of Abbreviations CMOS Complementary metal-oxide-semiconductor filed effect transistor. CMP Chemical mechanical polishing. ELOG Epitaxial lateral overgrowth. FCA Free carrier absorption fab Fabrication facility. GOI Germanium on insulator. IC Integrated circuit. MOSFET Metal-oxide-field-effect transistor. QW Quantum well. QD Quantum dot. RC Resistance-capacitance. SOI Silicon on insulator. SHR Shockley-Read-Hall TD Threading dislocation. TCAD technology computer aided design 0D,1D,2D Zero, one, and two-dimensional.      xix  Acknowledgements Firstly, I would like to express my sincere gratitude to my research supervisor Prof. Xia for the continuous support of my master study and related research. Her guidance has helped me in all the phases of research and writing of this thesis. I could not imagine having a better supervisor and mentor for my master study. I would also like to thank Crosslight Software Inc. for supporting the work in this thesis. I wish to express my sincere thanks to UBC Materials Engineering for providing me with all the necessary facilities for the research.  I appreciate the help from my friends and colleagues at UBC. Special thanks are given to Weijun Luo and Yiheng Lin, who taught me the operation of Raman measurements and X-ray diffraction and gave me useful suggestions on my project. Feiyang Cai, Guangnan Zhou and Yunlong Zhao were my group mates, and we had many interesting discussions during group meetings that greatly broadened my knowledge in microelectronics and photonics.  And finally, I wish to express my gratitude to my parents for their ever-lasting love and support, without which I cannot be here. I always owe them for their unconditional love. xx Dedication    To my father and my mother.     1 Chapter 1: Introduction 1.1 Mooreโ€™s law and the interconnect bottleneck Silicon (Si)-based microelectronics has been the engine of modern information technology for the last five decades. From the observation of Gordon Moore, the number of transistors on an integrated circuit (IC) doubles every two years because of the scaling down of gate length, junction depth, and gate dielectric thickness of metal-oxide-field-effect transistors (MOSFETs). This is known as Mooreโ€™s law [1]. As a result, the performance of electronic devices improved dramatically in the past decades.  However, such scaling has reached its physical limitation in recent years. From 2004 to 2010, the scaling factor of gate length is 0.9 per year [2], which is larger than the predicted scaling factor of 0.7 [1].  At the same time, the scaling down of the transistors does not always bring benefit. Observed from Figure 1-1, with the scaling of transistors, the delay of metal interconnect increases dramatically, but the intrinsic gate delay does not decrease that much. As a result, the RC delay caused by the metal interconnects becomes the bottleneck of computation speed and is known as the interconnect bottleneck. Although many new materials like low-K dielectric materials and new architectures have been applied to reduce the RC delay [3], they are not going to solve the problem in the long run. Moreover, the high power consumption caused by the metal interconnects is the main source of IC heating problems. The solution to these problems is to use optical interconnects to replace metal interconnect, especially for long distance communication.  2  Figure 1-1 RC delay time of metal wire and intrinsic gate delay vs. technology node [4].   1.2 Optical interconnects, Si photonics, and Si-compatible lasers Optical interconnects have been used in communications for decades especially in the long distance communication using optical fibers. There are several advantages of optical interconnects compared to metal interconnects. The first advantage lies in the low power consumption of optical interconnects. The main reason for the low power consumption is the zero rest energy of photons. As a result, it requires less energy to transmit photons [3]. Also, the attenuation of signals in the optical fibers is significantly lower than metal interconnects in long distance. The second advantage is the high speed and bandwidth. Since photons are immune to RC delay, the computation speed is not limited by the communication speed. Because photons at different wavelengths do not interact with each other, the wavelength-division multiplexing (WDM) technology can be applied in the multi-channel communication. Light signals at different wavelength can be transmitted simultaneously and thus increase bandwidth. Thirdly, the cost of silica fiber is lower than metal wire [3, 5].   3  With the successful application of optical interconnects in long distance, optical interconnects were gradually applied to a shorter distance like the rack-to-rack (1-100 m), board-to-board               (0.5-1 m), chip-to-chip level (1-50 cm) or on-chip (< 1 cm) levels. Short-reach optical communications (< 100 m) are widely used in data centres and supercomputers. For short-reach optical interconnects, more and more Si photonic technologies have been used. An example is Intelโ€™s optical ThunderboltTM cables that allow the optical connection of external peripherals to a computer [6-8].  Si photonics is an emerging technology to leverage the silicon CMOS fabrication technology for the creation of photonic devices. It has been identified as a path toward addressing the challenges of increasing functionality and bandwidth requirement of cloud computing and data centres. The low cost and high performance of integrated photonics circuits may meet the need of the continuing expansion of the internet infrastructures with higher and higher performance requirement. Figure 1-2 shows the main photonic devices in Si photonics. Si photonic systems require light sources, modulators, waveguides, and photodetectors. A modulator transfers the electric signals into optical signals, then the optical signals transmit through a waveguide and are captured by the photodetector and then be transferred back to electric signals. These devices are often connected to Si-based electronic circuits such as laser driver circuits and photodetector amplification circuits. All components, except for the light source, have been developed on a Si platform with high bandwidth capacity.  4  Figure 1-2 Key devices in silicon photonics [9]. There are two ways to implement the light source: one is to use external laser source and the other is to build an on-chip laser source. Although external laser sources with high emitting efficiency and good temperature stability can be used, they often suffer from problems like large coupling losses, design complexity, large footprint and high packaging expense [10]. In comparison, the on-chip laser sources have bigger potential to reach a higher integration density, better energy efficiency and better energy proportionality than external laser sources [10]. However, the development of on-chip laser sources has met great challenges. Due to the indirect band gap nature of Si, the Si-based laser sources have poor performance. Integrating traditional light emitting III-V semiconductors like GaAs or InP also encounter material compatibility issues like significant big mismatch in lattice constants and thermal expansion coefficients [11, 12]. Approaches to address these issues are discussed in Section 1.3. 1.3 Types of Si-compatible lasers In the past few decades, researchers all over the world have invested extensive efforts in finding solutions to a Si-compatible laser system. Now, several approaches have been proposed. Silicon based lasers like Si Raman laser [13], porous Si laser [14], Si nanocrystals [15] and Er-doped silicon laser [16-19] suffer from problems like intrinsic optical pumping mechanism, low optical  5 gain or limitation of Er concentration etc., which prevent these lasers from useful applications as efficient light source [10].  Integrating III-V materials, e.g. GaAs and InP, with Si platform has attracted much attention. Three main integration approaches have been extensively explored: direct mounting integration, wafer bonding based heterogeneous integration and direct hetero-epitaxial growth [10]. The direct mounting approach integrates a laser diode die with a Si on insulator (SOI) wafer directly with solder bumps. This method retains the superior characteristic of III-V laser diodes but suffers from inefficient end coupling and poor alignment precision. A sub-micron precision is required, and the facet reflection causes troubles to the whole system [10]. As a result, direct mounting can only be used as a temporary solution to the integration problem. The second integration method is through wafer bonding. III-V material is vertically coupled to silicon chips by wafer bonding. This approach is not subject to lattice mismatch and can be aligned with high precision. However, the wafer bonding method has disadvantages of high cost, low yield, and low integration density, and suffers from the wafer size mismatch [10]. The final integration method is the direct hetero-epitaxial growth of III-V materials on Si. Resulting from the large lattice constant and thermal expansion coefficient mismatch between Si and III-V materials, high density of threading dislocations (TD) occurs during the growth, which fails the optical devices. TD reducing technologies such as epitaxial lateral overgrowth (ELOG) along with a thick buffer layer are required to lower the TD density. Quantum wells (QWs) have been successfully grown on the ELOG InP layers with good optical properties [20]. Another way to overcome the problems caused by TD is to use nanostructures like quantum dots (QDs). The discrete distribution of QDs improves the defect tolerance. The direct hetero-epitaxial growth of InAs/GaAs QDs on Si has been realized 6 in recent years [21]. However, due to contamination issues, there is still a long way for III-V semiconductors to be adopted in the mainstream Si fabrication facilities (fabs).  Until now, companies have put lots of efforts in integrating III-V lasers on silicon and great progress has been achieved. In 2016, Intel release the first commercial 100 Gbps optical transceiver that integrate III-V lasers on silicon chip by wafer bonding. A start up company Skorpios also achieved their CMOS tunable laser in a similar way. Another company Luxtera also achieved their 100 Gbps optical transceiver by direct mounting integration. Ge-on-Si lasers are another competitive solution because they are fully compatible with the complementary metal-oxide-semiconductor field effect transistor (CMOS) technology, which will greatly reduce the process complexity, cost, and time to enter the fabs [10].  Optically pumped Ge laser was first realized in 2010 [22] and electrically pumped Ge lasers were demonstrated in 2012 [23] and 2015 [24] accordingly. However, the Ge lasers achieved have large threshold current and very low efficiencies. Debugging the poor performance and providing predictions of Ge laser potential are in great need. Although there are theoretical studies or calculations on Ge lasers, most of them are over simplified zero-dimensional (0D) or one-dimensional (1D) calculations and are not based on experimental data [25]. Experimental-data-based two-dimensional (2D) modeling and simulations were discussed only in one study with a fixed strain level [26]. As a result, it is important to come up with a new design and modeling to address the problem of low efficiency of Ge laser.  1.4 Structure of this thesis In this work, we proposed new designs to improve the efficiency and lower the threshold current of Ge laser. 2D laser modeling was performed and calibrated with experimental results. Optimizations of Ge lasers were conducted to maximize the laser performance.  7 The thesis is organized as below: In Chapter 2, we will review the progress and techniques in Ge lasers and define the problem. In Chapter 3, we will introduce a general laser theory to better understand how Ge lasers are modeled. In Chapter 4, we will discuss the Ge laser modeling, strain introduction and strain calculation. In Chapter 5, we will discuss the optimization results of Ge lasers and propose ways to improve Ge laser performance.  8 Chapter 2: Literature Review and Problem Definition 2.1 Band gap engineering methods Ge and Si are both Group IV semiconductors, and Ge is the most Si-compatible semiconductor.  They have the same crystalline structure (diamond cubic structure), a similar lattice constant and physical properties such as self-diffusivity, Youngโ€™s modulus, and Poisson ratio. Due to Geโ€™s unique optical properties and compatibility with Si, it plays an important role in the light sensing and modulation in the silicon photonics [27, 28]. However, Ge is an indirect band gap semiconductor as Si, which is inferior in light emitting applications than direct band gap semiconductors, such as Gallium Arsenide (GaAs) and Indium Phosphide (InP). Figure 2-1 (a) illustrates the band structure of Ge [29]. The band structure of Ge is called indirect band gap because the minimum energy state of the conduction band does not align with the maximum energy state of the valence band in K (wave vector) space. When an electron in L valley wants to recombine with a hole in the valence band, a third particle, normally a phonon, is needed to maintain the conservation of momentum because photons carry 0 crystal momentum. The involvement of phonons makes this process much less likely to occur, which makes the indirect band gap transitions in Ge 4 to 5 orders of magnitude less effective than that of direct band gap transmissions [30]. Fortunately, the direct band gap at the ฮ“ valley is only 136 meV higher than the indirect band gap at the L valley at room temperature. Therefore, by band gap engineering, it is possible to modify the band structure of Ge and turn Ge into a direct or pseudo-direct band gap material for efficient light emitting. Three methods have been proposed to modify the band structure of Ge.  9  Figure 2-1 Band structures of Ge under different conditions. (a) bulk Ge without strain or doping; (b) tensile strained Ge without doping; (c) highly n-type doped Ge without strain. Figure courtesy of Donguk Nam at Stanford University [29]. The first method to engineer the band structure of Ge is by introducing tensile strain. In 1996, Fischetti and Laux from IBM first presented the theoretical work that under 1.75% biaxial strain, Ge turned into a direct band gap material [31]. As shown in Figure 2-1 (b), with tensile strain, the conduction band edge is lowered, and the valence band is lifted up. The ฮ“ valley (direct) of conduction band approaches the light hole (lh) band top faster than the L valley (indirect), and under more than 1.75% biaxial or 4.6% uniaxial tensile strain, Ge becomes a direct band gap material [29]. It has been demonstrated that introducing small strain in Ge layer is beneficial for Ge-based optoelectronic devices like detectors and modulators. Moreover, tensile strains cause a light hole (lh) and heavy hole (hh) band splitting, which reduces the density of states near the band edge and provides an extra benefit of achieving a low threshold laser [10]. However, high strains also cause a redshift in the lasing wavelength from the technologically important 1550 nm wavelength range to longer wavelengths [10]. The redshift may be a problem for optical fiber communication since the 1550 nm L-band was selected according to the low loss windows of optical fibers. For on-chip interconnects, there are no such concerns. Due to the difficulties of introducing high-level tensile strains and constructing laser structures at the same time, no Ge 10 lasers have been built solely by strain engineering yet. Methods of obtaining high tensile strain will be discussed in Section 2.3. The second method, n-type doping (usually with phosphorus), is used to help achieve pseudo-direct band gap Ge. The difference between direct band gap and the pseudo-direct band gap is that the latter still has an indirect band gap structure, but the indirect band gap is filled up by doping to an energy level that is equal to the energy level of the lowest energy level of the direct band. Therefore, extra electrons in the indirect band have higher possibility to jump into the direct band  via intervalley scattering process [5]. Electrons in both bands can participate in recombinations, but the efficiency of recombinations from the direct band is much higher than that of the indirect band. In terms of recombination, the indirect band electron contribution can be safely ignored. However, the indirect band still has energy states that are in the same energy range as the direct band energy states as seen in Figure 2-1 (c).  As a result, electrons still need to fill the energy states in the indirect band (L valley) and the electrons going to the direct band (ฮ“ valley) are reduced. Although the pseudo-direct band gap is not ideal, it was shown to enhance the optical gain in Ge significantly, which made Ge lasers possible [32]. However, high doping also introduced a high free carrier absorption, and as a result, increased the threshold current greatly [5].  Tensile strain and high n-type doping are usually used together to achieve a pseudo-direct band gap condition. According to calculation results, an n-type doping concentration in the order of   1019 cm-3 with a 0.25% biaxial tensile strain can turn Ge into a pseudo-direct band gap material [32]. The first electrically pumped Ge laser was achieved by this method. The third method to modify the band structure of Ge is to grow a GeSn alloy. It was predicted theoretically that when Sn reached a certain concentration, GeSn alloy became a direct band gap material [33]. However, such concentration is not well studied, and in different theoretical models,  11 the value varies from 6% to 20% [3]. The key in achieving direct band gap by GeSn alloying is to achieve high-quality GeSn alloy with sufficient Sn concentration. The main challenge is the low solubility of Sn in Ge (<1%), which is far away from the required concentration [10]. As a result, problems like Sn precipitation, amorphization occur. Despite these difficulties, high-quality GeSn alloy with Sn concentration of 12.6% was achieved using chemical vapor deposition (CVD), and lasing was observed [34]. However, the GeSn laser demonstrated were optically pumped instead of electronically pumped with a lasing wavelength of 2.3 ยตm, and could only operate below 90 K [34]. GeSn is an interesting material for future CMOS compatible mid-infrared laser applications, but more researches are still needed.   Figure 2-2 Calculation and experimental result of band crossing of GeSn alloy as a function of Sn concentration. Figure courtesy of Huo Yijie at Stanford University [3].  2.2 Doping technologies A high doping in Ge is required in achieving pseudo-direct band gap structure. Ion implantation is the most common method to introduce dopants in semiconductors. By applying a high electric 12 field, ions are accelerated to obtain a high energy before shooting into the host semiconductors. The advantages of doping by ion implantation is its precision in controlling doping doses and its low processing temperature.  Although high n-type doping concentrations can be obtained by ion implantation, the implant damages, such as vacancy aggregation and material amorphization caused by the heavy ion collisions, are difficult to be resolved. These defects will become non-radiative recombination centres for carriers and reduce the direct band gap transition efficiency of the material. The results of the Kimerlingโ€™s group showed that the photoluminescence intensities of the Ge samples doped with ion implantation were 5 times lower than those of Ge samples doped with in-situ methods [5]. Photoluminescence is an important measurement of the direct band gap transmission efficiency. In theory, n-type doping can fill the L-level, which helps to achieve direct band gap transitions in pseudo-direct or indirect band gap semiconductors to enhance photoluminescence intensity. However, ion implantation also results in greater implant damage, which can counteract the effect of n-type doping [36]. As a result, Kimerlingโ€™s group and others research groups have argued that ion implantation may not be a good choice for n-type doping of Ge due to the fact that the damage caused by implantation is difficult to be completely removed.  In order to avoid implantation defects in materials, a variety of in-situ doping methods have attracted the attention of researchers. If gaseous phosphorus is used as the dopant source, and Reduced Pressure Chemical Vapor Deposition (RPCVD) and Ultra-High Vacuum Chemical Vapor Deposition (UHVCVD) are used for the doping, the maximum n-type doping concentration achievable is 1 ร— 19 cm-3 [37]. The reason why the doping concentration cannot increase lies under the contradiction between the phosphorus out-diffusion and low temperature requirement for high-quality materials growth [5]. To solve this problem, the Kimerlingโ€™s group proposed multi-layer  13 delta-doping method [38]. Several mono-layers of phosphorus on Ge (delta layers) were grown on top of the active region as the dopant source. The dopants then diffused into Ge by a drive-in annealing. Using such multiple stacked germanium-phosphorus layers as a dopant source on a 600 - 800 nm thick Ge film, a uniform n-type doping concentration of  4.0 ร— 1019 cm-3 was reached, which is the highest doping level achieved using a CVD method [38]. Doping concentration currently used in Ge laser is quite high and almost reaching the solubility limit of dopants. Further increase in the n-doping concentration will meet difficulties in controlling the dopant out-diffusion and growing high-quality Ge films simultaneously [10]. Among the three band gap engineering methods discussed in Section 2.1, introducing tensile strain is a more promising solution to improve the efficiency of Ge laser. 2.3 Stress engineering techniques for Ge As discussed, n-type doping is a double-bladed sword. It helps the band gap engineering, but at the same time increases the optical loss greatly. Stress engineering does not introduce the high optical loss. Several CMOS compatible methods have been applied to introduce strain in Ge. Thermal annealing is the most common and convenient strategy to do so. When Ge cools down from the growth temperature, typically 750 ยฐC, the thermal expansion coefficient mismatch between Ge and Si causes Ge to shrink more than Si and thus results in a tensile strain in Ge. However, the tensile strain introduced by this method is limited to 0.20% because of the strain relaxation above 750ยฐC [39].  By using backside silicidation of C54-TiSi2, which is a relatively simple method, the tensile strain in the front side Ge epitaxial layer has been increased from 0.20% of the 800 ยฐC as-grown sample to about 0.24% [39]. Other CMOS compatible approaches to achieve high strain levels in Ge include strain transfer from other intrinsic stressors and micromechanical engineering technology. 14 A common stress-introducting material in CMOS industry is silicon nitride (SiN). The stress level in SiN can be easily tuned to either compressive or tensile stress by changing the deposition conditions. The built-in stress in SiN has a wide range from -2 to +2 GPa [40]. When SiN is used as a compressive stressor, it releases stress by expanding and forcing the materials in contact to expand whereas a tensile stressor tends to cause a contraction in the adjacent areas. Therefore, In order to introduce a tensile strain, a compressive SiN stressor should be used as a top stressor to stretch the Ge layer, and a tensile SiN stressor should be used as a side stressor to pull the Ge layer from both sides. However, introducing strain merely using SiN is limited because of the low transfer efficiency resulting from the large thickness of the substrate [10]. Some micromechanical engineering methods are usually adopted. Capellini et al. used โ€œIโ€ and โ€œHโ€ shape structure and used silicon nitride layer as a top stressor to reach about 0.7% effective biaxial strain in Ge as shown in Figure 2-3 [41]. Jinendra et al. used SiN with intrinsic tensile stress of 1 GPa as the side stressors and etch away the buried oxide layer below the Ge layer to form a suspended platform and achieved  0.99% uniaxial strain and 0.82% biaxial strain experimentally [42].   15  Figure 2-3 Top SiN on Ge stripes for strain introduction  [41]. Figure reproduced with the permission from Optical Society of America. Ge microbridges like those in Figure 2-4 were designed to obtain high tensile stain by reducing the contacting area and enhancing the strain transfer efficiency [10]. By etching the oxide below, Ge microbridges were suspended and pulled to reach a 3.1% uniaxial tensile strain [43]. By transforming the epitaxially grown Ge onto oxide through wafer bounding and then etching away the oxide, the uniaxial strain in the Ge microbridge was further increased to 5.7% and a direct band gap Ge was experimentally achieved [44].  16  Figure 2-4 Suspended Ge microbridges with high tensile strain  [44]. Figure reproduced with the permission from OSA Photonics Research. 2.4 Progress of Ge lasers  In 2007,  Liu et al. from Prof. Kimerlingโ€™s group at MIT proposed that Ge can become a pseudo-direct band gap material by adding tensile strain and doping [32]. In 2010, they successfully fabricated a tensile-strained, n-type doped optically pumped Fabry-Perot Ge laser, which was the first Ge laser in the world [22]. A 0.25% biaxial tensile strain resulted from the thermal expansion mismatch between Si and Ge was achieved, and the doping concentration was about 1.3ร— 1019 to   2.5 ร— 1019 cm-3. When the injected carrier concentration reached 5.0 ร— 1019 cm-3, an optical gain of more than 60 cm-1 was observed, and the lasing wavelengths were between 1590 to 1610 nm.  In 2012, Camacho-Aguilera et al. demonstrated the first electrically pumped Ge-on-Si laser with 4 ร— 1019 cmโˆ’3 n-type doped Ge and 0.25% biaxial tensile strain, as shown in Figure 2-5. The lasing wavelengths were between 1520 nm and 1700 nm depending on the cavity thickness. A high  17 optical gain of over 400 cm-1 was observed in electrically pumped lasers to overcome the high loss resulted from the metal contacts and free carrier absorption in Ge and polysilicon. The threshold current density was 280 kA/cm2 [23].   Figure 2-5 Prototype of the electrically pumped Ge-on-Si laser device [23]. Figure reproduced with the permission from Optical Society of America. In 2015, Koerner et al. from the University of Stuttgart observed lasing from highly doped            (3 ร— 1019 cm-1) unstrained Ge [24]. However, the threshold current density of 510 kA/cm2 is higher than what was achieved by Kimerlingโ€™s group at MIT because there is no strain involved. Nevertheless, Kernerner et al.'s experimental results confirmed the reliability of the MIT findings. Other types of Ge laser are also under development. Lasing from GeSn alloy grown on Si was observed in 2014 [34] by optical pumping. No electrically pumped GeSn lasers have been demonstrated due to the fabrication difficulties. Ge quantum dots (QDs) laser formed by partially amorphizing through Ge-ion bombardment [35] was demonstrated recently, which shows the potential of Ge as a lasing material on Si.   2.5 Thesis Objective Although the strain introduction methods discussed above can generate high tensile strain inside Ge, they are not suitable for laser designs. The microstructures in [43, 44] are usually difficult to achieve, and it is very inconvenient to inject current and build a p-n junction on such structures. ~102 nm18 For the SiN/Ge stripes studied in [41], the top nitride covers the entire Ge top surface, making it difficult to implement a cladding layer and a top contact. The first goal of this work is to design a much more practical stress introduction method and a Fabry-Perot laser structure using this method. Details of the design will be shown in Section 4.6. This design does not interfere with laser structure implementation as the previous work in  [41, 43, 44].  A note needs to be added for choosing double-heterostructure Fabry-Perot laser structure here. Double-heterostructure Fabry-Perot lasers were widely used before the introduction of quantum well (QW) lasers. So far, forming QW Ge lasers has been difficult. Yan Cai et al. [45] performed a careful study on forming QW Ge lasers. As the intention of Ge lasers is to solve the material integration challenges between III-V semiconductors and Si, it is not reasonable to introduce        III-V semiconductors such as InAs on GaSb to form QWs on Ge. The only two options are forming SiGe/Ge/SiGe or relaxed Ge/tensile Ge/relaxed Ge QWs. Cai et al.โ€™s study showed that the former approach is not effective to reduce the threshold current (Ith) [45]. The latter approach, the relaxed Ge/tensile Ge/relaxed Ge structures are challenging to implement since achieving high strain in a thin layer is difficult, so there have not been any experimental efforts on that yet. Therefore, current Ge lasers are still double-heterostructure based.  The second goal of this work is to study whether the new design we proposed would be beneficial for Ge laser performance and to what extent we can optimize the structures to maximize the Ge laser performance. These two questions are important to address, as the available Ge stress engineering methods have major problems for laser structure designs and implementation. Since Ge lasers are still a new concept, it is important for the research community to know whether this topic has a potential with a practical design.    19 Chapter 3: General Laser Theories 3.1 Introduction of Fabry-Perot lasers and rate equation Double heterostructure lasers have a three-layer-sandwich structure with wide band gap, low index materials as the outer layer and a narrow band gap, high index materials in the centre as the active region. In this way, both carriers and photons can be confined in the central active region as shown in Figure 3-1. The outer layers are p and n-type doped respectively for the purpose of carrier injection, and the active region is usually undoped to reduce the photon absorption caused by the dopants. However, for our structures, the active Ge region is n-type doped to enhance the emission efficiency. The following theories discussed are called rate equation theory for lasers. Quantum mechanical calculations using the perturbation theory were used to calculate the optical transmission. Due to the page limit, only rate equation theory is discussed here.  Figure 3-1 Band diagram of a forward biased double-heterostructure laser diode [46]. In undoped or slightly doped active regions, the electron density (N) approximately equals to the hole density (P) because of the charge neutrality. Thus we can simplify the analysis by 20 specifically tracking only electron density, N, and use it as the carrier density. The change in carrier density N equals to the generation of carriers by injection minus the recombination of carriers, which reads:  ๐‘‘๐‘๐‘‘๐‘ก= ๐บ๐‘”๐‘’๐‘› โˆ’ ๐‘…๐‘Ÿ๐‘’๐‘ (3-1) Here, Ggen is the carrier generation rate, and Rrec is the rate of carrier recombination rate per unit volume in the active region. Ggen is related to the carrier injection and is determined by the injection current I multiplied with an injection efficiency ฮทi and then divided by electron charge q and active region volume Vol as shown in Eq. (3-2). The injection efficiency or internal efficiency ฮทi is defined as  the percentage of electrons and holes transporting into the active region after surviving the carrier losses due to recombination outside the active regions or carrier overflow into the other side of the p-n junction [47].  ๐บ๐‘”๐‘’๐‘› =๐œ‚๐‘–๐ผ๐‘ž๐‘‰๐‘œ๐‘™ (3-2)  ๐‘…๐‘Ÿ๐‘’๐‘ = ๐‘…๐‘†๐‘…๐ป + ๐‘…๐ด๐‘ข๐‘” + ๐‘…๐‘ ๐‘ + ๐‘…๐‘ ๐‘ก = ๐‘…๐‘’๐‘(๐‘) + ๐‘…๐‘ ๐‘ก  (3-3) The recombination rate Rrec is the combination of several effects: spontaneous recombination rate Rsp, SRH recombination rate RSRH, Auger recombination RAug,  and stimulated recombination rate, Rst as shown in Eq (3-3). Rsp, RSRH, RAug are all related to carrier density, so a carrier density related recombination term Rec(N) is used as shown in Eq. (3-3). Details of carrier loss are discussed in Section. 3.4. Rst is usually written as an individual term as ๐‘…๐‘ ๐‘ก = ๐‘ฃ๐‘”๐‘”(๐‘)๐‘†(๐‘ก) because it is related to the photon density S(t). vg is the group velocity, and g(N) is the gain coefficient. Rst represents the photon-stimulated net electron-hole recombination which generates more photons [48].   21 Taking Eq. (3-3) into Eq. (3-1), the rate equation for the carriers is re-expressed as:  ๐‘‘๐‘(๐‘ก)๐‘‘๐‘ก=๐œ‚๐‘–๐ผ๐‘ž๐‘‰๐‘œ๐‘™โˆ’ ๐‘…๐‘’๐‘(๐‘) โˆ’ ๐‘ฃ๐‘”๐‘”(๐‘)๐‘†(๐‘ก) (3-4) The rate equation for photons in Eq. (3-5) also consists of a loss and a generation term. Light oscillates in the laser structure by reflecting between two mirrors to trigger stimulated emissions. The light that travels inside the cavity can be either absorbed by the material or amplified by triggering the stimulated emission. The main photon generation term for the laser is the stimulated emission term Rst and the spontaneous emission term Rsp is ignorable. The gain that associates with Rst will be discussed in Section 3.5. As indicated in Figure 3-1, the cavity volume occupied by photons and carriers are different. As a result, the photon generation rate should be ๐‘‰๐‘œ๐‘™๐‘‰๐‘œ๐‘™๐‘๐‘…๐‘ ๐‘ก not just Rst [48]. The overlap coefficient ๐›ค =๐‘‰๐‘œ๐‘™๐‘‰๐‘œ๐‘™๐‘ is called the confinement factor, where Vol is the effective active volume for carriers and Volp is the active region for photons. ฮ“ is calculated as the ratio between the integration of light intensity in the active region and the integration of light intensity in all region [49].  ๐‘‘๐‘†(๐‘ก)๐‘‘๐‘ก= ๐›ค๐‘ฃ๐‘”๐‘”(๐‘)๐‘†(๐‘ก) โˆ’๐‘†(๐‘ก)๐œ๐‘โ„Ž๐‘œ 1/๐œ๐‘โ„Ž๐‘œ = (< ๐›ผ๐‘– > +๐›ผ๐‘š)๐‘ฃ๐‘” (3-5) The photons are lost through the absorption inside the cavity and the light emission at the mirror, which are represented by an internal loss coefficient ฮฑi and a mirror loss coefficient ฮฑm. The internal loss is usually expressed as <ฮฑi>, which is the weighted average of the internal loss over all region. We can also use a photon lifetime ฯ„pho to characterize the decay of photons as in Eq. (3-5). The loss mechanism of photons will be further discussed in Section 3.3.  22 3.2 Threshold condition When the carrier density is below the lasing threshold condition, the optical loss in the laser exceeds the gain, and the laser does not lase. As the carrier density increases to the point where the gain equals to the optical loss, rate equations reach the threshold condition [11]:  ๐›ค๐‘”(๐‘๐‘กโ„Ž) =< ๐›ผ๐‘– > +๐›ผ๐‘š (3-6) When the threshold condition in Eq. (3-6) is met, the carrier concentration N clamps at a threshold carrier concentration Nth. The gain g also clamps at a threshold gain gth. If the current injection increases, the carrier concentration in the cavity increases, and so does the gain g. As a result, more carriers are consumed through stimulated light emissions, which in return decrease the carrier density and gain. The excess carriers injected will recombine to generate photons by stimulated emissions [49]. Therefore, the carrier concentration and gain will clamp at the threshold values.  From the carrier aspect, the carrier absorption also equals to carrier generation since no stimulated emission happens yet. Therefore, threshold current Ith is calculated as in Eq. (3-7). It means that the carrier absorption (right-hand side) is equal to the carrier generation (left-hand side).  ๐œ‚๐‘–๐ผ๐‘กโ„Ž๐‘ž๐‘‘๐‘Š๐ฟ= ๐‘…๐‘ ๐‘Ÿโ„Ž(๐‘๐‘กโ„Ž, ๐‘ƒ๐‘กโ„Ž) + ๐‘…๐‘ ๐‘(๐‘๐‘กโ„Ž, ๐‘ƒ๐‘กโ„Ž) + ๐‘…๐ด๐‘ข๐‘”(๐‘๐‘กโ„Ž, ๐‘ƒ๐‘กโ„Ž) (3-7) The absorption terms on the right-hand side are electron and hole density dependent. Here d, W, L are the thickness, width, and length of the cavity.  ๐‘…๐‘ ๐‘(๐‘๐‘กโ„Ž, ๐‘ƒ๐‘กโ„Ž) , ๐‘…๐‘ ๐‘Ÿโ„Ž(๐‘๐‘กโ„Ž, ๐‘ƒ๐‘กโ„Ž) , and ๐‘…๐ด๐‘ข๐‘”(๐‘๐‘กโ„Ž, ๐‘ƒ๐‘กโ„Ž)  depend on both threshold hole density Pth and threshold electron density Nth. This equation holds true for doped region.    23  After the threshold condition being met, the excess carriers contribute to the stimulated emission. Therfore, the photon density at steady states can be calculated by [48]:  ๐‘† = ๐œ‚๐‘–๐ผ โˆ’ ๐ผ๐‘กโ„Ž๐‘ž ๐‘‰๐‘œ๐‘™ ๐‘ฃ๐‘”๐‘”๐‘กโ„Ž (3-8) The stored optical energy in the cavity can be calculated as: ๐ธ๐‘œ๐‘  = ๐‘†โ„๐œ”๐‘‰๐‘œ๐‘™๐‘ .The optical power output Pout is obtained by multiplying the Eos with the energy loss rate through the mirrors ๐‘ฃ๐‘”๐›ผ๐‘š [48]:  ๐‘ƒ๐‘œ๐‘ข๐‘ก = ๐‘ฃ๐‘”๐›ผ๐‘š๐‘†โ„๐œ”๐‘‰๐‘œ๐‘™๐‘ = ๐œ‚๐‘–โ„๐œ”๐‘ž๐›ผ๐‘š๐›ผ๐‘– + ๐›ผ๐‘š (๐ผ โˆ’ ๐ผ๐‘กโ„Ž) (3-9) If we define slope efficiency or the differential quantum efficiency ฮทd as in Eq.(3-10), Eq.(3-9) can be simplified as Eq. (3-11). After the threshold, the output power increases linearly with the injection current.  ๐œ‚๐‘‘ =ฮ”๐‘ƒ๐‘œ๐‘ข๐‘กฮ”๐ผโ„๐œ”๐‘žโ„ = ๐œ‚๐‘–๐›ผ๐‘š< ๐›ผ๐‘– > +๐›ผ๐‘š= ๐œ‚๐‘–๐œ‚๐‘’๐‘ฅ๐‘ก (3-10)  ๐‘ƒ๐‘œ๐‘ข๐‘ก =โ„๐œ”๐‘ž๐œ‚๐‘‘(๐ผ โˆ’ ๐ผ๐‘กโ„Ž)                                               (3-11)  3.3 Optical loss mechanisms The optical loss mechanisms in lasers are categorized in two types: photons can either be absorbed or escape from the cavity through the end facets for external use.  The loss resulted from the escape of photon at end facets is defined as the mirror loss ฮฑm and is expressed as:  ๐›ผ๐‘š =1๐ฟln (1โˆš๐‘…1๐‘…2)  (3-12) 24 L is the length of the cavity, R1, R2 are the reflectivity of two end facets. The mirror is usually achieved by simple end clipping or Bragg Grating. At a constant temperature, the loss of the mirror remains constant since the end-mirror reflectivity of the device does not change.  The loss resulted from the absorption is defined as internal loss ฮฑi. One source of the light absorption comes from the metal contact. Metal can absorb light greatly. As a result, when the metal contact is not well positioned, the metal contact can cause a serious light absorption, and cause high internal loss. Details will be discussed in Section 5.2.1.  Another source of internal loss inside highly doped semiconductor is the free carrier absorption (FCA) [50]. Free carrier absorption occurs when the free electrons or holes absorb photons and are excited to another excited state. Unlike interband absorption, the free electrons and holes that absorb photons are already excited. Free carrier absorption consists of two parts, free electron absorption (caused by free electrons from the n-type dopants and injected electrons) and free hole absorption (caused by free holes from the p-type dopants and injected holes). The FCA loss of a semiconductor are usually described by the Drude model [51]:  ๐›ผ = ๐ด๐‘๐œ†2 + ๐ต๐‘ƒ๐œ†2 (3-13) A and B are the free electron and free hole absorption coefficient, N and P are the electron and hole concentration in the unit of cm-3, ฮป is the light wavelength in the unit of nm. It can be seen from the formula that the free carrier absorption is proportional to the free carrier concentration. However, when we use the classical Drude model to describe the free electron absorption of biaxially strained germanium, which is in proportion to the square of the wavelength, the Drude model was found to be consistent with the experimental data only in the wavelength range of            ฮป > 15 ฮผm [52]. At ฮป < 15 ฮผm, the free carrier absorption decreases faster with wavelength than predicted by the Drude model [52]. Instead, the free electron absorption properties of Ge can be  25 described more accurately by using the first principle calculation of the intra-L-valley absorption [53] as compared to the Drude model. With a doping level of  4 ร— 1019 cm-3 and 0.25% tensile strain, the FCA loss is ~20 cmโˆ’1 in the wavelength range around 1550 nm, which is one order smaller than ~200 cmโˆ’1  calculated from the Drude model [52]. Compared with the 30 cm-1 FCA absorption in GaAs under 5.4 ร— 1018 n-type doping [54], Ge is slightly better in terms of optical loss.  3.4 Loss mechanism for carriers Electrons in the conduction band can eventually lose their energy and jump back to the valence band to recombine with holes. This process is called recombination. The energy released from the recombination is transferred in the form of photons or phonons heat. The recombinations that cannot generate photons for lasing are considered as losses. There are three types of recombinations that are considered as loss terms: 1) Spontaneous radiative recombination; 2) Shockley-Read-Hall recombination; 3) Auger recombination. 3.4.1 Spontaneous radiative recombination In radiative recombination, two forms of emission exist: stimulated emission and spontaneous emission. In spontaneous emission, electrons in the direct conduction band recombine with holes in the valence band and emit photons in random directions. Light emission in light emitting diode (LED) is a typical example of spontaneous emission in semiconductor devices. The recombination of electron-hole pairs can also be stimulated by incoming photons, which is called stimulated emission. The incoming photons interact with the electrons in the conduction band, causing it to 26 drop to valence band and emit photons with the same frequency, phase, direction and polarization as the incident photons. In a laser, photons emitted are created by the stimulated emission. The spontaneous emission is considered as a loss of carriers, but it can be ignored because most of the radiative emissions are consist of stimulated emissions. 3.4.2 SRH recombination SRH (Shockley-Read-Hall) recombination is also called trap-assisted recombination. Electrons transit from the conduction band to valence band through defects. Two steps are involved in SRH recombination. In the first step, electrons are trapped by the defect states in the forbidden band. These defects can be either unintentionally introduced to the materials, such as dislocation during fabrication, or intentionally added, like the dopants in the materials. In the second step, if a hole is also trapped at the same energy state as the previous electron, they recombine [55].  The recombination rate depends on the energy levels in the forbidden band introduced by the defects. If the defectsโ€™ energy level is close to either conduction or valence band, the carriers trapped have more possibility to be re-emitted to the conduction or valence band. Therefore, defects with mid-gap energy levels are very effective for SRH recombination. The SRH recombination rate is calculated by [56, 57]:  ๐‘…๐‘›๐‘ก๐‘— = ๐‘๐‘›๐‘—๐‘๐‘๐‘ก๐‘—(1 โˆ’ ๐‘“๐‘ก๐‘—) โˆ’ ๐‘๐‘›๐‘—๐‘1๐‘—๐‘๐‘ก๐‘—๐‘“๐‘ก๐‘— (3-14)  ๐‘…๐‘๐‘ก๐‘— = ๐‘๐‘๐‘—๐‘ƒ๐‘๐‘ก๐‘—๐‘“๐‘ก๐‘— โˆ’ ๐‘๐‘๐‘—๐‘ƒ1๐‘—๐‘๐‘ก๐‘—(1 โˆ’ ๐‘“๐‘ก๐‘—) (3-15) ๐‘…๐‘›๐‘ก๐‘—,๐‘…๐‘๐‘ก๐‘— are the electron and hole recombination rates per unit volume through jth deep trap level. Ntj is the density of the jth deep trap. cnj, cpj are the electron and hole capture coefficients of the jth deep trap. N is the electron density and P is the hole concentration. ftj is the occupancy possibility of the jth deep trap level, which is calculated from the Fermi distribution function. N1j is the electron concentration when the electron quasi-Fermi level coincides with the energy level Etj of the jth trap.  27 A similar definition applies to P1j.  The first term of Eq. (3-14) (3-15) represents the capture of electrons/holes by a recombination centre, the second term represents the rate of emission of electrons/holes from centres back to conduction/valence band. ฯ„nj and ฯ„pj are the electron and hole lifetime due to the jth trap, which are the average time for the carriers to decay. They indicate the quality of the material. The better quality the material has, the smaller the lifetime is, the fewer carriers are lost through defects. The lifetime is expressed with capture coefficients by the following relationship [57]:  1๐œ๐‘›๐‘—= ๐‘๐‘›๐‘—๐‘๐‘ก๐‘— (3-16)  1๐œ๐‘๐‘—= ๐‘๐‘๐‘—๐‘๐‘ก๐‘— (3-17) The capture coefficients cnj and cpj for electrons and holes due to the jth recombination centre can be further expressed as [57]:  ๐‘๐‘›๐‘— = ๐œŽ๐‘›๐‘—?ฬ…?๐‘› (3-18)  ?ฬ…?๐‘› = โˆš8๐‘˜๐‘‡๐œ‹๐‘š๐‘› (3-19)  ๐‘๐‘๐‘— = ๐œŽ๐‘›๐‘—?ฬ…?๐‘ (3-20)  ?ฬ…?๐‘ = โˆš8๐‘˜๐‘‡๐œ‹๐‘š๐‘ (3-21) ?ฬ…?๐‘›, ?ฬ…?๐‘ are the average thermal velocity of electrons and holes. ฯƒnj, ฯƒpj are the electron and hole capture cross sections of the jth deep trap. mn, mp are the bulk effective masses for electrons and holes. 3.4.3 Auger recombination Auger recombination is a three-particle interaction with 4 energy states. A typical type of Auger  recombination, CCCH Auger recombination, is shown in Figure 3-2. In Figure 3-2, when an 28 electron from energy state 1 jumps to state 2 to recombine with a hole, instead of releasing energy by emitting a photon, the energy is given to another electron to jump from state 3 to 4. This electron in state 4 then thermalizes back down to the conduction band edge. The main type of Auger recombination in Ge can be classified as CCCH, CCHS, and CHHS, where C stands for conduction band, H for valence band and S for the split-off band. The second and third character represent the original states of the particle, those being CC, CH, and HH; the first and fourth character represent the final state after the interaction, those being CH and CS, being CH and CS [30].  Figure 3-2 Diagram of CCCH (a) direct and (b) indirect Auger recombination. Figure courtesy of Rodolfo E. Camacho-Aguilera at MIT [30]. The Auger recombination rate is given by [47]:  RAug = (๐ถ๐‘›๐‘ + ๐ถ๐‘๐‘ƒ)(๐‘๐‘ƒ โˆ’ ๐‘๐‘–2)   (3-22) Cn and Cp are the electron and hole Auger recombination coefficient respectively. N and P are the electron and hole density while Ni is the intrinsic carrier density without doping. From the equation, we can see that the Auger recombination rate is proportional to the cube of carrier concentration under carrier injection condition. Therefore, under high carrier injection, Auger recombination becomes one of the main sources of non-radiative recombination. The typical value of Auger coefficient for Ge is in the order of 10-31 cm6s-1 [58]. However, from the research of Camacho-Aguilera [30], at high doping, the Auger coefficient should be in  29 the order of 10-32 cm6s-1, which is one order smaller than predicted by the established theory. This result is supported by the experimental results in [59] as shown in Figure 3-3. In the established theory, the radiative carrier lifetime ฯ„ in Ge should decrease linearly for both n-type and p-type dopants as illustrated in Figure 3-3 when doping concentration is smaller than 1018 cm-3. The radiative carrier lifetime represents the decay of carriers through radiative recombination. Therfore, the reduction of the radiative carrier lifetime means that the radiative recombination is reduced. However, the decrease in the radiative carrier lifetime is reduced under high doping concentration (>1018 cm-3). At a doping concentration around 1019 cm-3, the radiative carrier lifetime is around l0-6 s, which predicts an Auger coefficient in the order of 10-32 cm6s-1.    Figure 3-3 Radiative carrier lifetime of n-type and p-type dopants in Ge. Figure courtesy of Rodolfo E. Camacho-Aguilera at MIT [59] [30].  The smaller Auger coefficient can be explained by the indirect CCCH recombination. The threshold energies for different types of Auger recombination mechanisms in Ge were calculated 30 by Camacho-Aguilera [30] and are listed in Table 3-1. The indirect CCCH shown in                           Figure 3-2 (b) has the lowest threshold energy, which means a higher possibility of occurring. From Figure 3-2 (b), we can see that when CCCH indirect recombination happens, the carriers that originally stays in indirect L valley jump to direct ฮ“ valley. As a result, CCCH indirect recombination contributes to a higher electron concentration in ฮ“ valley and enhance the radiative emission instead of decreasing the radiative emission as predicted in established theory. Under high doping condition with doping level above 1018 cm-3, the increase of carrier concentration in ฮ“ band due to CCCH indirect recombination becomes prominent since Auger recombination increase proportional to the square of doping concentration. As a result, Auger recombination under high doping does not decrease the possibility of radiative recombination, and the Auger coefficient should be in the order of 10-32 cm6s-1 [59]. Compared with the typical Auger recombination coefficients in the order of  10-31 cm6s-1 for III-V materials [60], Ge is slightly better in terms of Auger recombination because of the CCCH indirect recombination. Table 3-1 Threshold energy for different Auger recombinations. Table courtesy of Rodolfo E. Camacho-Aguilera at MIT [30].   Heavy Hole interaction(eV) Light Hole interaction(eV) CCCH direct 0.87 1.12 CCCH L-indirect 0.2 0.03 CCCH ฮ”-indirect 0.24 0.036 CHHS 0.56 1.44 CHHL 0.82 0.82 3.5 Gain Gain is used to describe the optical amplification in the semiconductor laser. The gain we talk about in the thesis is the material gain, which is only related to the band gap and carrier density  31 and is not related to the geometry of the laser. When we consider the stimulated emission between energy level 1 and 2, the gain is related to the net stimulated rate [48]:  ๐‘…๐‘ ๐‘ก = ๐‘…21 โˆ’ ๐‘…12 = ๐œˆ๐‘”๐‘”S (3-23) Rst is the net stimulated rate, which is the difference between the stimulated emission rate from energy level 2 to 1, R21, and the stimulated absorption of photons from energy level 1 to 2, R12. ฮฝg is the group velocity, and S is the photon density. Writing out the Fermi factors explicitly, the two radiative transmissions R12 and R21 become:   ๐‘…21 = ๐ต21๐‘“๐‘(1 โˆ’ ๐‘“๐‘ฃ)๐‘† (3-24)  ๐‘…12 = ๐ต12๐‘“๐‘ฃ(1 โˆ’ ๐‘“๐‘)๐‘† (3-25) The fc and fv are the electronโ€™s occupation probability at energy level 2 in the conduction band and level 1 in the valence band. B21 and B12 are the Einstein coefficients for stimulated emission and absorption. The physical meaning of fc(1-fv) is the possibility of an electron in the conduction band to occupy the Ec level, and an absence occurs at the Ev level in the valence band. Similar meaning applies to fv(1-fc). Under carrier injection, the electrons in the conduction band and valence band satisfy the non-equilibrium Fermi-Dirac distribution, represented by equations [47]:  ๐‘“๐‘ =11 + exp (๐ธ๐‘ โˆ’ ๐ธ๐‘“๐‘๐‘˜๐‘‡  ) (3-26)  ๐‘“๐‘ฃ =11 + exp (๐ธ๐‘ฃ โˆ’ ๐ธ๐‘“๐‘ฃ๐‘˜๐‘‡  ) (3-27) Efc and Efv are the quasi Fermi levels of electrons in the conduction and valence band, which are determined by the injection level and doping. Ev and Ec are the energy levels associated with the photon transition. The energy difference of Ev and Ec equals to the photon energy: ๐ธ๐‘ โˆ’ ๐ธ๐‘ฃ = โ„๐œ”.  32 From detailed calculation, the B21 is equal to B12 [47]. Therefore, gain is proportional to the difference of the two Fermi function: ๐‘” โˆ (๐‘“๐‘ โˆ’ ๐‘“๐‘ฃ) From the theoretical calculations using Fermiโ€™s golden rule, optical gain is expressed as [5, 47]:  ๐‘”(โ„๐œ”) = ๐ถ0 ๐œŒ๐‘Ÿ(โ„๐œ” โˆ’ ๐ธ๐‘”)(๐‘“๐‘ โˆ’ ๐‘“๐‘ฃ) ๐œŒ๐‘Ÿ(โ„๐œ” โˆ’ ๐ธ๐‘”) =12๐œ‹2(2๐‘š๐‘Ÿโˆ—โ„2)32โˆšโ„๐œ” โˆ’ ๐ธ๐‘”  ๐ถ0 =๐œ‹๐‘’2|๐‘๐‘๐‘ฃ|๐‘๐œ€0๐‘š02๐‘›๐œ” (3-28) Here, |pcv| is related to the transition matrix factor, n is the refractive index of the material, both of which are the physical properties of the semiconductor material. ๐œŒ๐‘Ÿ is the joint density of states of direct conduction band (ฮ“ valley) and valence band as a function of photon energy near the band edge. Eg is the band gap energy.  mr* is the reduced effective mass defined by: 1๐‘š๐‘Ÿโˆ— =1๐‘š๐‘โˆ— +1๐‘š๐‘ฃโˆ—  , where mc*and mv* are effective mass of direct conduction and valence band, respectively. The effective mass will be further discussed in Section 4.3. fc-fv is called the population inversion factor. Under thermal equilibrium state and low carrier injection condition, the population inversion factor is negative, thus the gain is negative, showing the property of light absorption. Under high carrier injection level, the population inversion factor is positive, which means that the number of electrons in the conduction band is larger than that in the valence band, and a positive gain will appear. To have a positive gain, we have fc-fv>0, which results in:  ๐‘’(๐ธ๐‘โˆ’๐ธ๐‘“๐‘)/๐‘˜๐‘‡ < ๐‘’(๐ธ๐‘ฃโˆ’๐ธ๐‘“๐‘ฃ)/๐‘˜๐‘‡ or  ๐ธ๐‘“๐‘ โˆ’ ๐ธ๐‘“๐‘ฃ > ๐ธ๐‘ โˆ’ ๐ธ๐‘ฃ = โ„๐œ” (3-29) This requirement is called Bernard-Duraffourg inversion condition. Therefore, a positive gain exist only at ๐ธ๐‘” < โ„๐œ” < ๐ธ๐‘“๐‘ โˆ’ ๐ธ๐‘“๐‘ฃ.  33 Chapter 4: Ge Laser Related Modeling and Parameters Developed by Crosslight Software, LASTIPTM (LASer Technology Integrated Program) is a powerful two-dimensional(2D) device simulation program designed to simulate the semiconductor laser. With well-established physical models, it provides users with a quantitative insight into various aspects of a semiconductor laser. Given the structural and material properties, it simulates the laser characteristics. LASTIPTM is one of the most widely used and well-recognized laser simulation tool in industry and academic since 1995. As a result, we use it as the tool of our laser simulation.  Apart from the general laser models, some specific considerations need to be taken into account for Ge laser modeling to include the doping, stress, and interdiffusion effects. The parameters for the models were also carefully selected or fitted to experimental data as discussed below. 4.1 Doping-induced Ge band gap changes  In addition to the tensile strain effect, doping is another factor that causes band gap narrowing. Band gap narrowing is a common phenomenon in n-type or p-type doped semiconductor materials such as silicon, germanium and gallium arsenide. However, few studies have been done on the band gap narrowing effect of Ge with n-type doping. Haas et al. observed the band gap narrowing of the direct and indirect band gap of Ge under N-doping conditions by measuring the infrared absorption and concluded that the direct and indirect band gap have almost the same change under the influence of doping [61]. Recently, researchers also observed a direct band gap narrowing effect with 38 meV reduction in ฮ“ band in 1020 cm-3 n-type doped Ge LEDs [62]. Jain and Roulston described the indirect band gap narrowing effect of Ge under high doping conditions using the empirical formula Eq.(4-1) [63]. 34  โˆ†๐ธ๐‘”๐ฟ = 8.67 (๐‘๐ท1018)13+ 8.14 (๐‘๐ท1018)14+ 4.31 (๐‘๐ท1018)12 (4-1) Here, ND is the n-type dopantsโ€™ concentration in the unit of cm-3.  However, since the fitting parameters in Eq.(4-1) are mostly derived at T = 80 K, this model overestimates the indirect band gap narrowing effect at room temperature. Direct band gap narrowing effect of Ge with n-type doping is not theoretically analyzed in the literature. Therefore, in the modeling of this work, we used a linear relationship between the band gap change with the doping level to deduce an empirical expression for the direct band-gap narrowing effect as proposed in [64]:  โˆ†๐ธ๐‘”๐ฟ = 0.013๐‘’๐‘‰ + 10โˆ’21๐‘’๐‘‰/๐‘๐‘šโˆ’3 โˆ™ ๐‘๐ท (4-2) As shown in Figure 4-1, the calculated results in Eq. (4-1) are not in agreement with the experimental data but results of Eq. (4-2) are in good agreement with the direct and indirect band gap narrowing observed experimentally by Haas [61]. Therefore, it is assumed that the direct and indirect band gap of Ge have the same change under the n-type doping [45]. When the n-type dopantsโ€™ concentration was 4.0 ร— 1019 cm-3, the direct and indirect band gap narrowing was       0.053 eV.  35 0204060801000 1 2 3 4 5Haas indirect  bandgap narrowingEq. (4-2)Haas direct bandgap narrowingEq. (4-1)Band gap narroring (eV)n-type dopant's concentration ND (1019 cm-3) Figure 4-1 n-type band gap narrowing effect of different models at 300 K.  4.2 Stress effect and models The band gap dependence of Ge on strain can be calculated by kยทp theory, Pikus-Bir Hamiltonian and Luttinger-Kohn models [47, 65-67]. The band edge shift can be seen as a combined effect of a hydrostatic strain component (ฮตxx+ฮตyy+ฮตzz) and a shear or uniaxial strain component (ฮตxx+ฮตyy-2ฮตzz) as listed in Eq. (4-3) to (4-6) [68-70]:   ๐ธ๐ถโˆ’โ„Žโ„Ž(๐‘˜ = 0) = ๐ธ๐‘” โˆ’ ๐›ฟ๐ธโ„Ž๐‘ฆ +12๐›ฟ๐ธ๐‘ โ„Ž (4-3)  ๐ธ๐ถโˆ’๐‘™โ„Ž(๐‘˜ = 0) = ๐ธ๐‘” โˆ’ ๐›ฟ๐ธโ„Ž๐‘ฆ โˆ’12๐›ฟ๐ธ๐‘ โ„Ž (4-4)  ๐›ฟ๐ธโ„Ž๐‘ฆ = โˆ’๐‘Ž(๐œ€๐‘ฅ๐‘ฅ + ๐œ€๐‘ฆ๐‘ฆ + ๐œ€๐‘ง๐‘ง) (4-5)  ๐›ฟ๐ธ๐‘ โ„Ž = โˆ’๐‘(๐œ€๐‘ฅ๐‘ฅ + ๐œ€๐‘ฆ๐‘ฆ โˆ’ 2๐œ€๐‘ง๐‘ง) (4-6)  36 The hydrostatic strain component results from a volume change and does not break the symmetry of the crystal. As a result, the hydrostatic strain component only shifts the band edge of the conduction and valence band with a hydrostatic deformation energy ๐›ฟ๐ธโ„Ž๐‘ฆ without breaking any band degeneracy [70]. The uniaxial strain component breaks the symmetry of a crystal, and as a result, causes the split of the light hole valence band (lh) and heavy hole valence band (hh) with a shear deformation energy ๐›ฟ๐ธ๐‘ โ„Ž [68, 69]. ฮตxx, ฮตyy, ฮตzz are the normal strain in x, y, z direction, while x and y are in plain direction and z is the direction that is normal to the plane. a and b are the hydrostatic deformation potential and shear (uniaxial) deformation potential, respectively. If we write band gap of different conduction band separately and take into account the spin-orbit coupling, the band gap energy values in strained Ge are calculated using the following equations [47]:  Eฮ“โˆ’hh = Eg + aฮ“(ฮตxx + ฮตyy + ฮตzz) + Qฮต (4-7)  Eฮ“โˆ’lh = Eg + aฮ“(ฮตxx + ฮตyy + ฮตzz) โˆ’12(Qฮต โˆ’ ฮ” + โˆšฮ”2 + 2ฮ”Qฮต + 9Qฮต2) (4-8)  ELโˆ’hh = Eg + aL(ฮตxx + ฮตyy + ฮตzz) + Qฮต (4-9)  ELโˆ’lh = Eg + aL(ฮตxx + ฮตyy + ฮตzz) โˆ’12(Qฮต โˆ’ ฮ” + โˆšฮ”2 + 2ฮ”Qฮต + 9Qฮต2)          (4-10)  Qฮต = โˆ’b2(ฮตxx + ฮตyy โˆ’ 2ฮตzz) (4-11) Here, ฮ” is the spin-orbit splitting energy; aฮ“ and aL are hydrostatic deformation potential at ฮ“ and L valleys; Qฮต is the shear deformation energy. Eฮ“-hh, Eฮ“-lh are the direct band gap energies between the ฮ“ valley and the heavy hole and light hole band respectively. EL-hh, EL-lh are the indirect band gap energies between L valley and the heavy hole, light hole valence band, respectively. From these equations, we can see that heavy hole band is decoupled from the other bands, while light hole band is coupled with the split-off band [47]. Split-off hole (SOH) band is caused by the  37 spin-orbiting coupling with the band maximum point at K = 0 and is separated from lh and hh band with a spin-orbit splitting energy ฮ”. The SOH band is far away from the valence band top lh and hh bands. Therefore, it is usually ignored in the discussions of carrier recombination. However, the band shift of the lh band due to the coupling of lh and SOH band cannot be ignored because the resulted difference in lh energy is tens of meV, which is comparable to the lh and hh splitting [47].  The dependences of the band gap energies on biaxial strain are calculated according to Eq. (4-7) to Eq. (4-11) and shown in Figure 4-2 with the biaxial strain assumption that ฮตxx = ฮตyy, and             ๐œ€๐‘ง๐‘ง = โˆ’2๐ถ12/๐ถ11๐œ€๐‘ฅ๐‘ฅ. C11 and C12 are the elastic constants. The strain deformation potentials aฮ“, aL, ฮ” and b are obtained from [5, 68, 69] with the values of aฮ“ = -10.21 eV, aL = -4.02 eV,                     ฮ” = 0.296 eV  and b = -1.88 eV. As the biaxial strain increases, both the direct and indirect band gap decrease and the splitting of lh and hh band increases. The shrinkage of the direct band gap is more pronounced than that of the indirect band gap since the hydrostatic deformation potential for the indirect band aฮ“ is larger than aL. Ge turns into a direct band gap material at about 1.8% biaxial tensile strain, which is consistent with the literature work [71].  0.40.50.60.70.80 0.4 0.8 1.2 1.6 2EL-lhEL-hhE๏‡-lhE๏‡-hhBand gap (eV)In-Plane Strain (%) Figure 4-2 Direct and indirect band gap energies under different biaxial strain. 38 4.3 Effective mass of ฮ“ conduction band The E (energy)-K (wave vector) relationship of the free electron is ๐ธ(๐พ) =โ„2๐พ22๐‘š๐‘’, in which me is the free electron mass while โ„๐พ is the momentum. In a semiconductor crystal, electrons move in a periodic potential field of nuclei and the band edge of conduction and valence band can be approximated by a quadratic equation if we replace the free electron mass with an effective mass m*: ๐ธ(๐‘˜) =โ„2๐พ22๐‘šโˆ— [56]. We can obtain the effective mass from the second derivative of E-K diagram. In general, the effective mass might be anisotropic and is a tensorial, which is defined as:  1๐‘š๐‘–๐‘—โˆ— โ‰ก1โ„2๐œ•2๐ธ(๐พ)๐œ•๐พ๐‘–๐œ•๐พ๐‘— (4-12) However, since the ฮ“ conduction band is isotropic, we only have one value mฮ“*.  A larger effective mass corresponds to a band with a smaller curvature in E-k relation. As a result, a larger effective mass can increase the density of states around the band edge as described in Eq. (3-28) in Section 3.5. Therefore, it can increase the material gain as shown in Figure 4-3, which is calculated from LASTIPTM. A larger effective mass also causes the quasi-Fermi level Efc to decrease because the conduction band is broader. The left intersection on the gain profile in Figure 4-3 represents the quasi Fermi level difference Efc-Efv as described in Eq. (3-28). We can see that the change of quasi Fermi level difference due to the change of effective mass is negligible. However, the density of states increases a lot. As a result, the gain increases.  39 0500100015001.3 1.4 1.5 1.6 1.7 1.8Material Gain(cm-1)๏ฌ๏€ ๏€จ๏ญm)m*=0.03m*=0.04m*=0.05m*=0.06 Figure 4-3 LASTIPTMโ€™s results of material gain at different mฮ“* with 0.25% biaxial strain carrier concentration n=p=4ร—1019cm-3. Although in theory, the effective mass of Geโ€™s gamma band should be a constant with a value of 0.041me [72], Si-Ge interdiffusion makes it a variable. Si-Ge interdiffusion is unavoidable during Ge layer growth, defect annealing and dopant activation annealing. Especially, such interdiffusion is much faster for n-type-doped Ge/Si. Figure 4-4 from [73] shows the Ge profile after defect annealing. More interdiffusion occurs in Phosphorus-doped-Ge/Phosphorus-doped-Si (PGPS) in comparison with undoped-Ge/undoped-Si (UGUS). The thermal budget was 725 โ„ƒ for 64 mins, which is comparable to the condition used in Kimerlingโ€™s group [23]. Interdiffusion changed part of Ge to a SiGe alloy region. If we define the interdiffusion region is where Si counts for 2% or more, then this region is 200 nm thick for a one-micron thick P-doped Ge/P-doped Si. For the Ge laser with 100 to 300 nm Ge layer thickness, we expect that the interdiffusion region counts for a large portion of the Ge active region, and cannot be ignored.  40  Figure 4-4 Ge profiles measured by SIMS. The Ge profiles are shifted laterally for easy comparison. UGUS: undoped-Ge/undoped-Si; UGPS: undoped-Ge/ Phosphorus-doped-Si; PGUS: Phosphorus-doped-Ge/undoped- Si; PGPS: Phosphorus-doped-Ge/Phosphorus-doped-Si. The black dash line is the Ge profile of sample PGPS before annealing. Figure courtesy of Feiyang Cai at UBC [74].     Due to the limitation of LASTIPTM and the unavailability of the calibrated Ge profile data in the work of Kimerlingโ€™s group, we were not able to include the full interdiffusion profile into the simulations. Considering the interdiffusion induced SiGe alloy region, it is reasonable to have a bigger effective mass than the Ge value of 0.041me because Siโ€™s effective mass at ฮ“ band (0.22me) is much larger than that of Ge. Compared to other parameters such as band gap, meฮ“* has a stronger dependence on the Si molar fraction in the intermixed region due to such a large effective mass difference between Si and Ge. Moreover,  meฮ“* also have a big impact on the L-I behavior as shown in  Figure 4-6 on page 47. Therefore, we used the effective mass as a fitting parameter to take the Si-Ge interdiffusion into account.    41 4.4 Figures of merits of lasers Several parameters are used for the characterization of laserโ€™s performance. These parameters include threshold current Ith, slope efficiency ฮทd, internal efficiency ฮทi, extraction efficiency ฮทext, maximum wall-plug efficiency ฮทwp, confinement factor, internal loss, etc.  4.4.1 Threshold current Threshold current Ith is an important parameter to evaluate semiconductor lasersโ€™ performance. The smaller the threshold current is, the easier it is for the laser to lase. The magnitude of Ith can be derived from the intersection of L-I curve on the I axis.  We chose to use Ith instead of threshold current density Jth as the optimization criteria because Ith can reflect the actual laser performance. Ith relies heavily on the size of the cavity as described in Eq.(3-7). The longer or wider cavity requires more injection current than a smaller cavity. The structures we discuss have a similar cavity structure, the double-heterostructure Fabry-Perot laser, with changes only in geometry. As a result, using Ith for comparison can reflect the actual performance of the device. In the following discussion, we use Ith for comparison. Detailed discussions about Ith have been done in Section 3.2, and the expression of Ith is listed in Eq. (3-7). 4.4.2 Slope efficiency Slope efficiency ฮทd is defined in Eq.(3-10). ฮทd is related to the slope of the L-I (Light-Current) curve (ฮ”Pout/ฮ”I) of the laser. It represents the efficiency of photon generation from injected electron-hole pairs. Similar to the desire to lower threshold current of the device, the slope efficiency should be as high as possible when designing the semiconductor laser. A higher ฮทd means that a laser can generate higher output power in a smaller increment in the current. Under ideal conditions, the value of ฮทd is 100%. In a hypothetical ideal laser, over the threshold, an electron-hole pair injected recombines and emit one photon. The released photon resonates in 42 the optical cavity and is finally emitted from the laser. However, in practical circumstances, only part of the electron-hole pairs can be released through radiative recombination. Only a ratio of the total carriers injected can stay in the active region, which is represented by ฮทi. The rest of the electron-hole pairs might recombine outside the active layers, pass through the active region without recombination. The carriers inside the cavity still partly lost through non-emitting recombination, which is represented by the threshold current. Then, not all of the released photons can be successfully emitted from the cavity. Part of the photons generated gets absorbed by defects or in the optical cavity as discussed in Section 3.3. The ratio of photons coupling out of the cavity to the total photon density is defined as the extraction efficiency ฮทext. It is calculated as the ratio of mirror loss to the total loss in the cavity as in Eq. (3-10). The internal loss always exists, which indicates that ฮทext is smaller than 1 and not all photons can be emitted out. As a result, ฮทd is the product of internal efficiency ฮทi and extraction efficiency ฮทext to represents the electron-to-photon conversion efficiency. 4.4.3 Maximum wall-plug efficiency Wall-plug efficiency is the core parameter for the laser devices. It is defined as the ratio of optical power generated to the electrical power consumed and represents the overall energy conversion efficiency of the laser. The electrical power that is not converted to optical power is dissipated in the form of heat. Heat generation is one of the leading cause of laser degradation [75]. Therefore, a higher wall-plug efficiency does not only cut down the electric power needed for a certain optical output power but also lowers the power budget for the cooling system.  The voltage dependence of the current of the laser is similar to PN diode. When the current is above the threshold current, the diode is forward biased, and the laser can be seen as a resistor. The operation voltage Vop is operation current Iop multiplied by the series resistance R plus the  43 diode voltage VD as described in Eq (4-13). Therefore, the final expression for wall-plug efficiency is shown in Eq (4-14). In the calculation of ฮทwp, 2D laser L-I and I-V simulations were performed up to about 10 mW optical output, above which, to save computation time, L-I and I-V curves were extrapolated linearly up to about 200 mW optical output according to Eq. (4-13), (4-14). From Eq. (4-14), we can observe that wall-plug efficiency peaks and then decrease. We define the maximum wall-plug efficiency as ฮทwp, which is shown in Eq. (4-14), to evaluate the performance of laser because it represents the best energy conversion efficiency of the device.    ๐‘ƒ๐‘’๐‘™๐‘’๐‘๐‘ก = ๐ผ๐‘œ๐‘๐‘‰๐‘œ๐‘ = ๐ผ๐‘œ๐‘(๐‘‰๐ท + ๐‘…๐‘ ๐ผ๐‘œ๐‘) = ๐‘‰๐ท๐ผ๐‘œ๐‘ + ๐‘…๐‘ ๐ผ๐‘œ๐‘2                      (4-13)  ๐œ‚๐‘ค๐‘ = ๐‘€๐‘Ž๐‘ฅ[๐‘ƒ๐‘œ๐‘๐‘ก๐‘ƒ๐‘’๐‘™๐‘’๐‘๐‘ก] = ๐‘€๐‘Ž๐‘ฅ[โ„๐œ”๐‘ž๐œ‚๐‘‘(๐ผ๐‘œ๐‘โˆ’๐ผ๐‘กโ„Ž)๐‘‰๐ท๐ผ๐‘œ๐‘+๐‘…๐‘ ๐ผ๐‘œ๐‘2 ]                                           (4-14)  4.5 Modeling with LASTIPTM and model calibrations   We first simulated the experimental laser structure from Kimerlingโ€™s group [23] to calibrate our model. The cross section is illustrated in Figure 4-5. The Ge cavity was 1 ฮผm wide, and 270 ฮผm long, and the thickness of Ge active layer was set to be 200 nm, which was the average value of the 100~300 nm thickness in the experiments due to the process non-uniformity [23, 30]. On both left and right side of the Ge cavity, a 0.5 ฮผm wide oxide was used for optical confinement, and                     180 nm thick polysilicon was deposited on top of Ge as the cladding layer. The metal layers contained two layers of 0.1 ฮผm thick Titanium (Ti) and 1 ฮผm thick Aluminum (Al) layers as illustrated in Figure 4-5. 2 ฮผm Si substrate was used in the simulations. Virtual contacts were defined underneath the bottom of the Si substrate and on the top of metal layers for the biasing purpose, and they do not have interactions with light. This simplification is valid because the light field only penetrates Si to a depth much less than 2 ฮผm as shown in Figure 4-8 (c)-(e). The doping and the strain are the same as the experiments reported: Si substrate was 5 ร— 1019 cm-3 n-type doped. 44 Ge was 4 ร— 1019 cm-3 n-type doped with 0.25% biaxial tensile strain resulted from the thermal mismatch. Polysilicon is 5 ร— 1020 cm-3 p-type doped.  The strain-dependent Ge energy band gap model in Section 4.2 and the doping induced band gap narrowing effect in Section 4.1 were implemented in LASTIPTM. The metal-semiconductor heterojunctions were aligned by electron affinity as described in [47]. The reflectivity values of two facet are R1 = 23% and R2 = 38%, which correspond to a mirror loss ๐›ผm of 45 cm-1 [45]. Auger coefficients used were Cn = 3.0 ร— 10-32 cm6/s and Cp = 7.0 ร— 10-32 cm6/s [30, 32]. The index of refraction values of all materials were wavelength dependent. Since the surface is correctly passivated in the experiment, surface recombination can be ignored [30]. 1 ns of defect limited carrier lifetime (๐œ๐‘,๐‘› = 1 ๐‘›๐‘ ) was used as a conservative estimation [76]. This is a conservative assumption compared with the measurement of Geโ€™s carrier lifetime of 5.3 and 3.12 ns in recent studies [77, 78]. For simplicity, a default setting in the software with a uniform distribution of donor mid-gap traps with a density of 1010 mโˆ’3 was used. LASTIPTM takes ๐œ๐‘,๐‘› as an input to calculate the capture coefficients cnj, cpj according to Eq. (3-16)(3-17) and then SRH recombination rates ๐‘…๐‘›๐‘ก๐‘—, ๐‘…๐‘๐‘ก๐‘—are calculated accordingly by Eq. (3-14)(3-15).  For the optical loss, we assumed that the internal loss and the mirror loss are the primary sources of the loss and internal loss is dominated by the free carrier absorption [50]. In LASTIPTM, the free carrier absorption in a narrow wavelength range was simplified as ฮฑ=AN+BP. We used the maximum value of the first principle calculations results of free carrier absorption in n-type doped Ge for n-loss coefficient: ๐ด = 5.0 ร— 10โˆ’19  [52] , and the experimental measurement results in          p-type doped Ge [79] and the modeling results in [26] were used as a starting point to obtain the best fitting to the L-I curve in [23]. The effective mass of gamma conduction band (meฮ“*) was assumed to be equal everywhere in the Ge cavity and was used as the first fitting parameter of L- 45 I curve. The p-loss coefficient B was used as the second fitting parameter. The best fitting was obtain when meฮ“* = 0.0457 me and the best fitting free carrier loss relation was ๐›ผ๐‘– = 5.0 ร—10โˆ’19๐‘ + 1.023 ร— 10โˆ’17๐‘ƒ. The free carrier absorption properties of the silicon substrate and the polysilicon covering layer can be obtained from the conclusions in [80] and [81] as shown by equations (4-15) and  (4-16).  Free carrier absorption of the silicon substrate can be described as:  ๐›ผ = 1.8 ร— 10โˆ’18๐‘ + 2.7 ร— 10โˆ’18๐‘ƒ (4-15) Free carrier absorption of polycrystalline silicon cladding layer is:   ๐›ผ = 1.079 ร— 10โˆ’17๐‘ + 7.47 ร— 10โˆ’18๐‘ƒ  (4-16) We can observe that under high doping condition of our structure, polysilicon and silicon have higher loss than the Ge cavity. As a result, light that travels outside of Ge cavity experiences higher optical loss. The parameters used for simulation are summarized in Table 4-1. Using these parameters, our model produced a Jth of 300 kA/cm2 or Ith of 800 mA at 15 โ„ƒ with the transverse electric (TE) mode lasing at ฮป = 1676 nm. These results were very close to the experimental values of                              Jth = 280 kA/cm2 and the lasing wavelength of 1650 nm [23]. As seen in Figure 4-6, the model can match the experimental L-I curve quite well. Sensitivity test results are shown in Figure 4-6, which shows how a smaller FCA parameter or a smaller meฮ“* are not fitting the experimental data. We can observe that the simulation results are quite sensitive to the change of meฮ“*. A 1% change in meฮ“* will result in a 10% change in Ith.    46 Table 4-1 Material parameters used in the modeling of Ge laser [26] Material parameters Value Material parameters Value Cavity thickness 0.2 ฮผm Ge refractive index at 1550nm 4.18 Cavity width 1 ฮผm Si refractive index at 1550nm 3.46 Cavity length 270 ฮผm Max mobility for electrons 3900 cm2V-1s-1 Biaxial strain 0.25% Max mobility for holes 1900 cm2V-1s-1 Temperature 288 K Gamma band effective mass 0.0457me Facet reflectivity R1,R2 23%, 38% L band effective mass 0.22me Electron Auger coefficient Cn 3.0ร— 10-32 cm6s-1 Heavy hole effective mass  0.284me Hole Auger coefficient Cp 7.0ร— 10-32 cm6s-1 Light hole effective mass  0.043me  It should be noted that although Xiyue Li et al.โ€™s models [26] also fit the same experimental data, they used the FCA (free carrier absorption) coefficients as the fitting parameters and did not consider Si-Ge interdiffusion. In our modeling, we chose to use meฮ“* as the fitting parameters to include the effect of interdiffusion. One should notice that fitting meฮ“* to the experimental data is a rough estimation to take the interdiffusion into account and further study is required. The limitation of this method will be discussed in Section 4.7.  47  Figure 4-5 Cross-section of the Ge-on-Si heterojunction laser structure simulated.     Figure 4-6  L-I curves for the experimental result, calibration result, and sensitivity tests with a smaller FCA loss: ๐œถ๐’Š = ๐Ÿ“. ๐ŸŽ ร— ๐Ÿ๐ŸŽโˆ’๐Ÿ๐Ÿ—๐‘ต + ๐ŸŽ. ๐Ÿ—๐Ÿ๐Ÿ‘ ร— ๐Ÿ๐ŸŽโˆ’๐Ÿ๐Ÿ•๐‘ท, and a smaller effective mass meฮ“* = 0.0453 me.  4.6 Modeling of Ge laser with stressors We studied two structures with silicon nitride stressors as illustrated in Figure 4-7: structure 2 with side stressors only and structure 3 with top and side stressors together. The structure without stressors in Figure 4-6 (a) has the same structure as the electrically pumped Ge laser in  [23] and 00.511.5100 150 200 250 300 350Best fittingExperimental dataSmaller  FCASmaller effective massOptical Power (mW)J(kA/cm2)48 is named as structure 1 for comparison. In structure 2, we replace the oxide in structure 1 with side SiN stressors to introduce strain in Ge cavity. In structure 3, a top stressor was added along with the side stressors. The top stressor was put on top of the polysilicon layer to leave room for current injection. The top stressor has the same width as the Ge cavity and is 0.3 ฮผm thick. The consideration is that a wide c-SiN blocks current flow from the top contact to the substrate, which means an increase in series resistance. If the top stressor is too narrow, then the stress effect will not be as big. (a)   (b)   Figure 4-7 Laser structure simulated (cavity width = 1 ฮผm, thickness = 0.2 ฮผm, length = 270 ฮผm, cladding thickness = 0.18 ฮผm). (a) Structure 2: with side nitride stressors; (b) Structure 3: with top & side nitride stressors. The width of the top nitride stressor is the same as that of the Ge cavity. The metal contacts are composed of Ti and Al same as those in Ref. [23], shown in Figure 4-6. The refractive index of SiN is around 1.9 and is set to be wavelength dependent according to [82]. Therefore, it is suitable for optical confinement. The 2D optical field of 3 structures are shown in Figure 4-8 (c)-(e). We can observe that the SiN stressors can confine the light as good as the oxide. From the 1D cut profile in Figure 4-8 (a), we can observe that the light at the top is absorbed by the top metal contact in structure 1 and 2 and this absorption causes high optical loss. In structure 3,  the absorption caused by the metal contact is diminished because of the existence of SiN top stressor as shown in Figure 4-8 (b).  49 (a)  (b)  (c)    (d)   (e)   Figure 4-8 Normalized light intensity of 1D cut in the centre line of (a) structure 2 (structure 1 have a similar distribution); (b) structure 3. 2D profile of (c) structure 1; (d) structure 2; (e) structure 3. 50 We used the software TSUPREM-4TM for the modeling of strain field in structure 2 and 3. TSUPREM-4TM is a standard and extensively used two-dimensional (2D) technology computer aided design (TCAD) tool for process simulation in the semiconductor industry.  The advantage of TSUPREM-4TM is that it contains well-calibrated models based on decades of industry practice and includes several mechanisms for stress/strain introduction. These mechanisms include volume changes during oxidation, thermal mismatch between materials, intrinsic strain in deposited layers, and surface tension.  For strain introduced by SiN stressors, a plane strain assumption (strain in length direction         ฮตyy = 0) is suitable since the length dimension is much longer than the width and thickness dimensions. A tensile strain value of 0.25% in Ge resulted from the thermal expansion mismatch between Ge and Si in ฮตxx and ฮตyy was added on top of the strain introduced by stressors as the two types of strain comes from two independent sources. Geโ€™s Youngโ€™s modulus and Poisson ratio used were 102 GPa and 0.28 respectively. Silicon nitride has a Youngโ€™s modulus value ranging from 100 GPa to 350 GPa depending on the deposition method and recipe. For simplicity, a medium Youngโ€™s modulus value of 200 GPa in [83] was used for silicon nitride and the Poisson ratio used was 0.24. The intrinsic stress values used for tensile silicon nitride (t-SiN) and compressive silicon nitride (c-SiN) were +2 and -2 GPa respectively, both of which were common values already achieved in CMOS processing. After every deposition and etch step, a stress relaxation step was used in the software to calculate the stress change.  Since ฮตxx and ฮตyy appear in the form of ฮตxx+ฮตyy in the strain-dependent band gap models in Eq.(4-7) to (4-11) as discussed in Section 4.2, it is reasonable to use effective biaxial strain ๐œ€๐‘’๐‘ =(๐œ€๐‘ฅ๐‘ฅ+๐œ€๐‘ฆ๐‘ฆ)2  to represent the in-plane strain magnitude for the following discussions. The strain maps of ฮตeb and ฮตzz of structure 2 and 3 are illustrated in Figure 4-9 and Figure 4-10.   51  (a)  (b)  Figure 4-9 2D Strain map on the cross section: (a) ฮตeb (b) ฮตzz of structure 2 with side stressors only (cavity width = 1 ฮผm, thickness = 0.2 ฮผm, cladding thickness = 0.18 ฮผm).   52 (a)  (b)  Figure 4-10 2D Strain map on the cross section: (a) ฮตeb (b) ฮตzz of structure 3 with side and top stressors (cavity width = 1 ฮผm, thickness = 0.2 ฮผm, cladding thickness = 0.18 ฮผm).  53 Significant ฮตeb were introduced by the stressors as shown in the strain map. The average value in the centre line was used to represent the strain field for simplification. With only side stressors, 0.36% average ฮตeb was introduced including the 0.25% strain caused by the thermal expansion mismatch. The value was increased to 0.39% by adding a top stressor and we can see from the strain map that the ฮตeb in the upper part of Ge is greatly increased. This strain enhancement is not optimized due to the non-optimized Ge width and thickness as the stress introduction strongly depends on the sizes of cavity and stressors, and their relative positions.  The three average strain (ฮตxx, ฮตyy, and ฮตzz) in the centre line were loaded into LASTIPTM. Due to the limited capability of LASTIPTM in introducing strain, some transformation is needed. In LASTIPTM, A biaxial assumption (ฮตxx = ฮตyy = ฮตb, ฮตzz = โˆ’2C12ฮตxxC11 ) is used as the strain model, and only biaxial strain ฮตb is used as an input. However, from the strain simulation results, the strain gained from SiN stressors does not meet biaxial assumption. In order to incorporate non-biaxial strain into LASTIPTM, the effective deformation potential aeff and beff discussed below were used. Under the biaxial assumption, the hydrostatic term in Eq. (4-5) becomes:  ๐›ฟ๐ธโ„Ž๐‘ฆ = โˆ’๐‘Ž(๐œ€๐‘ฅ๐‘ฅ + ๐œ€๐‘ฆ๐‘ฆ + ๐œ€๐‘ง๐‘ง) = โˆ’๐‘Ž (2 โˆ’2๐ถ12๐ถ11) ๐œ€๐‘ (4-17) We use ฮฒ=ฮตzz/ฮตeb to express ฮตzz in terms of effective biaxial strain ฮตeb. Then the hydrostatic term becomes:  ๐›ฟ๐ธโ„Ž๐‘ฆ = โˆ’๐‘Ž(๐œ€๐‘ฅ๐‘ฅ + ๐œ€๐‘ฆ๐‘ฆ + ๐œ€๐‘ง๐‘ง) = โˆ’๐‘Ž(2 + ๐›ฝ)๐œ€e๐‘ = โˆ’๐‘Ž๐‘’๐‘“๐‘“ (2 โˆ’2๐ถ12๐ถ11) ๐œ€๐‘’๐‘ (4-18)  ๐‘Ž๐‘’๐‘“๐‘“ =๐‘Ž(2 + ๐›ฝ)(2 โˆ’2๐ถ12๐ถ11) (4-19) If we use aeff instead, the hydrostatic term is in the same form with the biaxial assumption. A similar approach was applied to uniaxial components: 54  ๐›ฟ๐ธ๐‘ โ„Ž = โˆ’๐‘(๐œ€๐‘ฅ๐‘ฅ + ๐œ€๐‘ฆ๐‘ฆ โˆ’ 2๐œ€๐‘ง๐‘ง) = โˆ’๐‘(2 โˆ’ 2๐›ฝ)๐œ€e๐‘ = โˆ’๐‘๐‘’๐‘“๐‘“ (2 +4๐ถ12๐ถ11) ๐œ€๐‘’๐‘ (4-20)   ๐‘๐‘’๐‘“๐‘“ =๐‘(1 โˆ’ ๐›ฝ)(1 +2๐ถ12๐ถ11) (4-21) After implementing the strain data into LASTIPTM using the method above, laser simulation of three structures were performed. The L-I curve of three structures are shown in Figure 4-11. Compared to Structure 1, by adding the side stressors, about 523 mA reduction in Ith and 1.05% increase in ฮทwp were obtained. The increased strain greatly improves the performance of laser. And in Structure 3, by adding the top and side stressors together, about 761 mA reduction in Ith and 12.9% increase in ฮทwp were obtained (Figure 4-11 and Table 4-2). The significant performance improvement introduced by the top stressors is because that top stressor not only introduces higher strain, but also decreases the optical loss caused by the metal contact as discussed previously.  Figure 4-11 L-I curve comparison for the three structures before the structure optimizations.   051015200 200 400 600 800 1000I (mA)Optical Power (mW) (3) w/ side & topstressors(2) w/ side stressors(1) w/o stressors 55 Table 4-2 Laser performances of the three structures in Figure 4-11. Structure 1 2 3 ฮตeb 0.25% 0.36% 0.39% Ith (mA) 810 287 49 Slope efficiency ฮทd 6.33% 7.20% 31.94% ฮทwp 2.07% 3.12% 16.03% 4.7 Limitations of our modeling Although our models can fit the experimental data quite well, it still has some uncertainties and limitations. The parameters that influence the results most are the effective mass, defect limited carrier lifetime, Auger coefficients, FCA coefficients, and strain. The uncertainties of these parameters lie in four aspects: 1) the not-well-studied Ge parameters, 2) modeling simplification and limitation of the simulation tool, 3) limited experimental data, and 4) the lack of research of the delta valley in Ge band structure.  Firstly, Ge is not a well-studied optical material, and many model parameters do not have widely agreed values or even ranges. The Auger coefficients discussed in Section 3.4.3 are a typical example. Under high doping and carrier injection, the values of Auger coefficients of Ge are one order smaller than that of lightly-doped Ge [30]. The FCA loss discussed in Section 3.3 is also one order smaller than predicted by the traditional Drude model [52]. For the defect limited carrier lifetime, it highly depends on the fabrication method, and no experimental results are available. As a result, there is a large uncertainty associated with the value of ฯ„p,n. The parameters we used mostly come from the research of Kimerlingโ€™s group as they pioneered the research of Ge optical properties, and provided a complete set of theoretical studies and experimental data on Ge lasers. Secondly, several simplifications were used in our modeling. Due to the limitation of LASTIPTM, only one strain value can be assigned to Ge. As a result, we used the average strain in the centre 56 line of Ge cavity and assumed that the strain in the Ge cavity is constant in the Ge cavity. From the strain field in Figure 4-9 and Figure 4-10, we can observe that most part of the cavity shares similar strain values with the central line and only the edges of the Ge cavity have a large value. Since the heterogeneity of strain only happens on the edges of the Ge cavity and light concentrates in the centre of the cavity, the strain in the centre is more important. Therefore, using the average strain in the centre line is reasonable. Another simplification is to use the effective mass of gamma conduction band, mฮ“*, as a fitting parameter to capture the Si-Ge interdiffusion since LASTIPTM is not able to load a real Ge concentration profile. Details were discussed in Section 4.3. In the structure optimization in Chapter 5, we assumed that mฮ“* does not change with the geometry. In reality, more complicated processes are involved. For example, the method used in the work of Kimerlingโ€™s group to fabricate highly doped Ge is a low temperature/high temperature (LT/HT) epitaxial growth with delta doping. It commonly has three major steps. The first step is the initial Ge seeding layer growth at low temperature (300 โ€“ 400 ยฐC). This layer is 50 - 100 nm thick and is highly defected due to the large lattice mismatch between Ge and Si that needs to be relaxed in this layer. The second step is a high-temperature growth (600 โ€“ 850 ยฐC) of Ge, which is of higher Ge film quality and faster growth rate. Therefore, we can see that if the Ge cavity is thicker, a bigger percentage will be grown by the HT step and the interdiffusion effect is weaker since the most interdiffusion region is the bottom Ge region (the seeding layer region). The third step is delta doping and followed by a high temperature (~ 700 ยฐC) and long time (~ 60 mins) drive-in annealing at which most of the interdiffusion happens. Multilayers of dopants were grown on top of active Ge layer and then a drive-in annealing is required for the dopants to diffuse into Ge cavity. A thicker cavity requires longer drive-in time for the dopants to diffuse over the whole cavity, and as a result, the  57 interdiffusion becomes more serious. Therefore, the interdiffusion region might account for the similar percentage in the Ge thickness and a constant mฮ“* is reasonable. The impact of interdiffusion on laser performance is an important topic for future studies since Ith is rather sensitive to the mฮ“*. However, all of these interdiffusion processes are difficult to model at this point due to the limited experimental data. The only experimental data is at a thickness around    0.2 ฮผm.  Another uncertainty lies in the delta valley of Ge, which is the second indirect valley in the conduction band that located only 50 meV above the ฮ“ valley. Under high injection condition, the delta valley might consume carriers and result in a higher threshold current. However, the research about the impact of delta valley on laser performance is very limited, and the band shift in the presence of strain is not well studied. In our modeling, we were not able to take the delta valley into account since LASTIPTM can only handle two conduction band valleys.  Even with these simplifications, our modeling is more realistic and physical than previous studies. Some theoretical works, like the study of Liu et al. [32] only studied the material parameters such as gain and loss. These works are important but too primitive to predict the performance of Ge lasers. The works in [25, 76] used simple rate equation models with lots of simplifications to calculate threshold current and slope efficiency. Such models were oversimplified and ignored lots of 2D features such as the carrier distribution, current leakage, light distribution, etc. Only the optical loss of Ge was considered, with the parasitic optical absorption in the metal contact ignored, which turned out to be an influencing factor as indicated in our modeling in Section 5. Although one 2D work was done in [84], it was based on an unrealistic structure that was difficult to fabricate, and the modeling was not calibrated with experimental data. Compared to these studies, our modeling is a great improvement. 58 Chapter 5: Structure and Strain Optimizations To take the full advantage from the stressors and further improve the device performance, we optimized the Ge cavity and the cladding geometry. We optimized W, dGe, and dpoly, which stand for the Ge cavity width, thickness, and the polysilicon cladding layer thickness respectively. The Ge cavity length was set to be unchanged at 270 ฮผm. Based on Xiyue Li et al.โ€™s study [26], geometry can greatly improve the efficiency. With the presence of SiN stressors, changing geometry also changes the strain field in the cavity and as a result influence the laser performance. Due to the limitations discussed in Section 4.7, in our optimization, we assume that the only geometry dependent material parameter is the strain.  In our optimization process, the goal is not to find the โ€œtrueโ€ optimal point, but rather to show that Ge lasers can be improved significantly. The reasons for that are twofold. 1) Ge is not a well-studied optical material, and many simplifications are assumed in our modeling as discussed in Section 4.7. Therefore, it is still too early to find the โ€œtrueโ€ optimum at this point. 2) Optimizing one variable at a time is more doable, as the rate equations are well established, and one can check the correctness of the results conveniently. We changed one parameter at a time and kept others constant. Optimizations were performed on three structures, and we will use structure 2 as a detailed example to illustrate the optimization process. Several parameters are used for the characterization of laser performance. These parameters include: Slope efficiency ฮทd, internal efficiency ฮทi, extraction efficiency ฮทext, Threshold current Ith, max wall-plug efficiency ฮทwp, confinement factor, internal loss, etc. For lasers, small Ith and large ฮทwp are both desired, but they may not be met at the same time. We chose maximum wall-plug efficiency ฮทwp as the most important optimization criteria because it represents the energy conversion efficiency of the device.  59 5.1 Optimizations of structure 1 without stressors For Structure 1, without the stressors, the Ith and ฮทwpโ€™ s dependence on W, dGe, and dpoly are shown in Figure 5-1 and Figure 5-2. Details will be discussed in Section 5.2. dpoly had the largest impact and was first optimized as in Figure 5-1. Ith decreases from 810 to 57 mA and ฮทwp increases from 2.07% to 20.8% when dpoly changes from 0.2 to over 0.8. We chose dpoly = 0.8 ฮผm as the optimized dpoly. For the W dependence, Ith increases linearly with W, but ฮทwp does not change much with W (Figure 5-2 (a)). Therefore, we chose W = 0.5 ฮผm as the optimized W for less Ith. For the dGe dependence, the increase of ฮทwp resulted from the better optical confinement cannot compete with the increase of Ith. As a result, the ฮทwp only increase from 18.5% to 23.5% and then decrease (Figure 5-2 (b)). We chose the peak point dGe =0.5 ฮผm as the optimization point for dGe dependence. The highest efficiency reached is 23.5% with dpoly = 0.8 ฮผm, W = 0.5 ฮผm๏ผŒdGe = 0.5 ฮผm, and          Ith = 63 mA.  Figure 5-1 Ith and ฮทwp of Structure 1, dpoly dependence. 0200400600800100005101520250.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Threshold current I th (mA)no StressorsW=1 ๏ญmdGe=0.2 ๏ญmPoly-Si Thickness dpoly (๏ญm)Wall-Plug efficiency ๏จwp (%)60  (a)  (b)  Figure 5-2 Ith and ฮทwp of Structure 1 (a) width dependence; (b) dGe dependence.  30405060708016182022240.5 0.6 0.7 0.8 0.9 1 1.1 1.2Threshold current I th (mA)Ge width  W (๏ญm)no Stressorsdpoly=0.8 ๏ญmdGe=0.2 ๏ญmWall-Plug efficiency ๏จwp (%)204060801001201401618202224260.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Threshold current I th (mA)Ge Thickness dGe (๏ญm)no stressorsdpoly=0.8 ๏ญmW=0.5 ๏ญmWall-Plug efficiency ๏จwp (%) 61 5.2 Optimizations of structure 2 with side stressors only 5.2.1 Polysilicon thickness (dpoly) optimization The results of polysilicon thickness dependence are shown in Figure 5-3. The polysilicon thickness dpoly has the most dominant effect in the geometry optimization. As dpoly increased, we observed a dramatic increase in ฮทwp and a decrease in Ith (Figure 5-3 (b)). This is because that the light absorption caused by the metal contact is greatly reduced. Since the difference in refractive index between germanium and polysilicon is small, the vertical optical confinement is poor. Therefore, when the polysilicon coating is thin, a considerable proportion of light will easily enter the metal contact area, resulting in serious light absorption. As the top metal contact moved further away from the Ge cavity with the increase in dpoly, the occurrence of light leakage is reduced, and the internal loss <ฮฑi> caused by the metal contact can be significantly reduced.  As a result, ฮทext and thus ฮทd increase monotonically and reach a plateau at thick dpoly                              (Figure 5-3 (a)). Ith decreases as dpoly increases since less carrier density is needed to compensate for the loss. As a consequence, ฮทwp increases to 21.3% and plateaus after dpoly = 0.8 ฮผm and Ith decreases to 54 mA (Figure 5-3 (b)). We chose 0.8 ฮผm as the optimization point since ฮทwp plateaued after that point.  62 (a)  (b)  Figure 5-3 Polysilicon thickness dpoly dependence (W =1 ฮผm, dGe = 0.2 ฮผm) of (a) <ฮฑi> and ฮทext ,ฮทd  (b) Ith and ฮทwp.  0123456510152025303540450.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1loss SlopeextractInternal loss <๏กi> (102cm-1)Efficiency (%)Poly-Si Thickness dpoly (๏ญm)Side stressors onlyW=1 ๏ญmdGe=0.2 ๏ญm๏จext๏จd5010015020025030005101520250.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Threshold current  Ith (mA)Wall-Plug efficiency ๏จwp (%)Side stressors onlyW=1 ๏ญmdGe=0.2 ๏ญmPoly-Si Thickness dpoly (๏ญm) 63 5.2.2 Ge Width (W) optimizations The cavity width (W) dependence is shown in Figure 5-4 and Figure 5-5 . ฮทwp only increases slightly and Ith increases linearly with W. (a)  (b)  Figure 5-4 Ge width W dependence (dpoly =0.8 ฮผm, dGe = 0.2 ฮผm) of (a) strain ฮตeb and ฮ“ (b) <ฮฑi> and ฮ“.  0.30.350.40.450.50.550.350.40.450.50.6 0.8 1 1.2 1.4Strain (%)Ge Width W (๏ญm)Confinement factor ๏‡Side stressors onlydpoly=0.8 ๏ญmdGe=0.2 ๏ญm7080901001100.350.40.450.50.6 0.8 1 1.2 1.4Internal loss <๏กi>(cm-1)Ge Width W(๏ญm)Side stressors onlydpoly=0.8 ๏ญmdGe=0.2 ๏ญmConfinement factor ๏‡64 (a)  (b)  Figure 5-5 Ge width W dependence (dpoly =0.8 ฮผm, dGe = 0.2 ฮผm) of  (a) ฮทd and ฮทext (b) Ith and ฮทwp.  28303234363840283032343638400.6 0.8 1 1.2 1.4Slope efficiency ๏จd (%)Ge Width W(๏ญm)Extraction efficiency ๏จext (%)Side stressors onlydpoly=0.8 ๏ญmdGe=0.2 ๏ญm2030405060708016182022240.6 0.8 1 1.2 1.4Ge Width  W (๏ญm)Side stressors onlydpoly=0.8 ๏ญmdGe=0.2 ๏ญmThreshold current  Ith (mA)Wall-Plug efficiency ๏จwp (%) 65  Figure 5-6 Impacts of Ge width (W) on other parameters. The cavity width W and thickness dGe dependence come from three different effects: 1) strain introduction, 2) optical confinement factor ฮ“ and 3) active region volume. The relationship between different parameters is shown in Figure 5-6. The tensile strain decreases with the increase of W because side stressors move away from the centre (Figure 5-4 (a)). The separation between the direct and indirect band gap increases accordingly, which results in a reduction in ฮทi. The decrease in strain raises the band gap, causing the lasing wavelength to become smaller. The reduced lasing wavelength causes a slight increase in the refractive index and thus increases ฮ“. As the cavity becomes wider, the lateral confinement becomes better, which also increases ฮ“       (Figure 5-4 (a)). The FCA loss of polysilicon is greater than Ge. Therefore, a bigger ฮ“ means less light travels in the lossy polysilicon region, which results in the decrease of <ฮฑi> (Figure 5-4 (b)) and the increase of ฮทext and thus the growth of ฮทd (Figure 5-5 (a)). Ith is a combination effect of nth, ฮทi, and geometry as indicated in Eq. (3-7), but mostly dominated by geometry since Ith increases almost linearly with W in Figure 5-5 (b). The wider the W is, the larger current is needed to compensate the carrier loss resulted mainly from Rsrh and RAug.  The increase of ฮทd would increase ฮทwp whereas increased Ith would decrease ฮทwp. Because of this competing effect, ฮทwp only increases slightly with W, as shown in Figure 5-5 (b). Further simulations show that choosing the maximum ฮทwp point where W = 1 ฮผm does not promise better 66 performance in d dependence since a narrower waveguide is desired for side stressors. On the contrary, a wider cavity increases Ith greatly. Therefore, we chose W = 0.5 ฮผm as the optimization point, where ฮทwp = 18.61% now but promotes the potential for higher efficiency. We choose W = 0.5 ฮผm as the minimum value for optimization because when W is too small, optical confinement in the horizontal direction becomes poor, which is not suitable for the laser and causes convergence problems in the simulation.  5.2.3 Ge thickness dGe optimization The cavity thickness (dGe) dependence is shown in Figure 5-7 to Figure 4-10. ฮทwp increases a lot and Ith increases monotonously with dGe.  Figure 5-7 Ge thickness dGe dependence (W =0.5 ฮผm, dpoly = 0.8 ฮผm) of Ith and ฮทwp.     283032343638404215202530350.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Threshold current I th (mA)Ge Thickness  dGe (๏ญm)Wall-Plug efficiency ๏จwp (%)Side stressors onlydpoly=0.8 ๏ญmW=0.5 ๏ญm 67 (a)  (b)  Figure 5-8 Ge thickness dGe dependence (W =0.5 ฮผm, dpoly = 0.8 ฮผm) of (a) Strain ฮตeb and ฮ“ (b) <ฮฑi> and ฮ“.   0.450.50.550.60.650.70.30.40.50.60.70.80.911.10.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Strain (%)Ge Thickness  dGe (๏ญm)Side stressors onlydpoly=0.8 ๏ญmW=0.5 ๏ญmConfinement factor ๏‡0204060801001200.30.40.50.60.70.80.910.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Ge Thickness  dGe (๏ญm)Side stressors onlydpoly=0.8 ๏ญmW=0.5 ๏ญmConfinement factor ๏‡Internal loss <๏กi>(cm-1)68  203040506070809020304050607080901000.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Ge Thickness dGe(๏ญm)Slope efficiency ๏จd (%)Extraction efficiency ๏จext (%)Side stressors onlydpoly=0.8 ๏ญmW=0.5 ๏ญm  Figure 5-9 Ge thickness dGe dependence (W =0.5 ฮผm, dpoly = 0.8 ฮผm) of  ฮทd and ฮทext.  Figure 5-10 Impacts of Ge thickness dGe on other parameters. The dependence of dGe is similar as Wโ€™s dependence, which is shown in Figure 5-10. Strain ฮตeb increases with dGe because more stressors react on the Ge cavity (Figure 5-8 (a)). The increase in the strain also caused a slight decrease in ฮ“ as discussed before.  ฮ“ increases with dGe since thicker cavity promote better vertical confinement (Figure 5-8 (a)). The ฮ“ shrinkage due to the change of lasing wavelength is only a minor effect for ฮ“. The increase in ฮ“ causes the <ฮฑi> to shrink                (Figure 5-8 (b))  and thus increases the ฮทext (Figure 5-9). As a result, ฮทd increases a lot since ฮทi and ฮทext are of the same trend (Figure 5-9). Same as the W dependence, Ith increases almost linearly  69 with dGe (Figure 5-7). As the competing effect of Ith and ฮทd, ฮทwp peaks at 34.8% and then decreases (Figure 5-7). We chose dGe = 0.8 ฮผm as the optimization point.  5.2.4 Strainโ€™s Impact If we make the stress in SiN stressors to be 0, the only difference between the structures with and without the stress of stressors is the strain inside the Ge cavity. As a result, by comparing the same structure in dGe dependence with and without the stress of stressors, we can see how strain influences the laser performance. For the structure with the stress of stressors, the strain inside Ge cavity increases monotonously with the increase in dGe as shown in Figure 5-11(a) while the strain keeps unchanged at 0.25% in the structure without the stress of stressors. The increased strain decreases the difference between n gamma (direct) and L (indirect) conduction band (not shown here), decreases the band gap and increases the separation between lh and hh band (Figure 5-11(b)). These changes in the band make the direct recombination easier, and as a result, increase the material gain (Figure 5-12(a)). With tensile strain engineering and high doping, a material gain over 2000 cm-1 can be achieved, which is quite close to the typical material gain value of III-V materials around 5000 cm-1 at lasing condition. In Figure 5-12 (a), we can observe that as the strain increase, the peaks of the gain have a red shift because of the shrinkage of the band gap. As the strain increase, a second peak, which results from the split of lh and hh valence band occurs. The increase in gain decreases the carrier density needed for lasing and thus reduce Ith  (Figure 5-12(b)). Ith decreases to about one-third with the stress of stressors, compared to that of the one without the stress of stressors. The increased lasing wavelength (Figure 5-11 (a)) decreases ฮ“ by the changed real index n and decreases ฮทext as discussed before (Figure 5-13 (a)). The ฮทd increases while ฮทext 70 decreases by the decreased ฮ“, which shows that the ฮทi increases with the strain for the same geometry (Figure 5-13 (b)).         (a) 0.450.50.550.60.650.70.751.751.81.851.91.9520.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Strain (%)Lasing wavelength ๏ฌ (๏ญm)Side stressors onlydPoly=0.8 ๏ญmW=0.5 ๏ญmGe Thickness  dGe(๏ญm)  (b)-1.5-1-0.500.511.50 0.5 1 1.5 2Energy (eV)Distance (๏ญm)n-type Gep-typePoly-Sin-type   SiBlue: 0.68% ๏ฅebRed: 0.25% ๏ฅeb๏„cb=15 meV๏„lh= -68 meV๏„hh= -35 meV Figure 5-11 (a) Strain ฮตeb  and ฮป with different dGe (W=0.5ฮผm, dpoly = 0.8ฮผm): (b) direct band alignment under different strain at thermal equilibrium (V=0, dGe=0.8 ฮผm).  71  (a)50010001500200025001.4 1.5 1.6 1.7 1.8 1.9 2Material Gain(cm-1)๏ฌ๏€ ๏€จ๏ญm)dGe=0.2๏ญm ๏ฅeb=0.49%dGe=0.4๏ญm ๏ฅeb=0.59%dGe=0.6๏ญm ๏ฅeb=0.66%dGe=0.8๏ญm ๏ฅeb=0.69%dGe=1๏ญm ๏ฅeb=0.71%  (b)  Figure 5-12 Strain impact with different dGe (W=0.5ฮผm, dpoly = 0.8ฮผm): (a) material gain at different strain with carrier concentration n=p=4ร—1019cm-3, (b) Ith, and (f) ฮทd.  204060801001201400.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Threshold current I th (mA)Side stressors onlydpoly=0.8 ๏ญmW=0.5 ๏ญm no stresswith stressGe Thickness  dGe(๏ญm)72  (a)  (b)  Figure 5-13 Strain impact with different dGe (W=0.5ฮผm, dpoly = 0.8ฮผm): (a) ฮทext (b) ฮทd. 304050607080901000.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Side stressors onlydpoly=0.8 ๏ญmW=0.5 ๏ญmno stresswith stressExtraction efficiency ๏จext (%)Ge Thickness  dGe(๏ญm)2530354045505560650.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Ge Thickness  dGe(๏ญm)Side stressors onlydpoly=0.8 ๏ญmW=0.5 ๏ญmno stresswith stressSlope efficiency ๏จd(%) 73 5.3 Optimizations of structure 3 with top and side stressors A large W and small dGe are desired for the strain introduction from the top stressor, which is undesired for the side stressors. Therefore, W and dGe are optimized together to obtain a high ฮทwp. W = 0.5 ฮผm is not the optimized width in Figure 5-14 (a), but by comparing a few different W values, W = 0.5 ฮผm has the potential to produce a higher ฮทwp. dGe = 1 ฮผm is the optimization point of dGe dependence for W = 0.5 ฮผm. For such structures, the strain introduction from side stressors is more prominent than the top stressor. The strain introduction from top stressor is not obvious because top stressor can only introduce strain on the top of the Ge cavity whereas side stressors can introduce strain in the whole cavity by decreasing the cavity width.  dpoly has similar but weaker impact compared to structure 2. Due to the presence of the top stressor, the top metal contact loss is greatly reduced for Structure 3 before increasing dpoly.  However, the increasing dpoly would further increase ฮทwp and decrease Ith, which shows that top stressor can only diminish the optical loss caused by metal to a certain extent. Increasing dpoly is a more efficient way to reduce the optical loss caused by the metal. The final optimization is: dpoly = 0.4 ฮผm, W = 0.5 ฮผm, dGe = 1 ฮผm, with ฮทwp peaks at 25.4% and an Ith of 44 mA. 74 (a)  (b)  Figure 5-14 Ith and ฮทwp  of Structure 3 (a) width dependence (b) dGe dependence. 3040506070801012141618200.6 0.8 1 1.2 1.4Thresdold current I th (mA)Ge Width W(๏ญm)Top+Side stressorsdPoly=0.18 ๏ญmdGe=0.2 ๏ญmWall-Plug efficiency ๏จwp (%)30354045501214161820222426280.2 0.4 0.6 0.8 1 1.2Ge Thickness dGe(๏ญm)Top+Side stressorsdpoly=0.18 ๏ญmW=0.5 ๏ญmThresdold current I th (mA)Wall-Plug efficiency ๏จwp (%) 75  Figure 5-15 Ith and ฮทwp  of Structure 3, dpoly dependence.  5.4 Comparisons of the structures The comparisons of the three structures after optimization are shown in Figure 5-16                                             and Table 5-1. We can observe that changing geometry could significantly increase ฮทwp and decrease Ith. Adding stressors can further improve these two parameters. By using side stressors only, the highest ฮทwp rose to 34.8%, but adding top stressor does not provide greater ฮทwp. This is mainly because that the top stressor increases the series resistance significantly. From the current density in Figure 5-17 and Figure 5-18, we can see that the top SiN stressor prevents the current from directly flowing through the cavity. Structure 1 and 2 have a series resistance around 0.4 ฮฉ, but it is around 0.9 ฮฉ for Structure 3, which means Structure 3 requires higher voltage and thus higher electric power. Plus, the strain introduced by top stressor is marginal compared to side stressors as discussed in Section 5.3. As a result, Structure 3 does not produce a higher ฮทwp than 4142434445462224262830320.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Top+Side stressors W=0.5 ๏ญmdGe=1 ๏ญmPoly-Si Thickness dpoly (๏ญm)Thresdold current I th (mA)Wall-Plug efficiency ๏จwp (%)76 Structure 2. Therefore, considering both ฮทwp and Ith, Structure 2, with side stressors only, is recommended. For cavity length dependence, we observe a linear relationship between Ith and length L as in Eq. (3-7). As a result, the ฮทwp increases as the L decreases because of the decline in Ith. A smaller cavity is desired to have low Ith and ฮทwp but might be limited by experimental problems like poor heat dissipation.  Figure 5-16 L-I curve for three structures after optimization. Table 5-1 Laser performance of the 3 structures in Figure 5-16 after structure optimizations. Structure 1 2 3 ฮตeb (%) 0.25 0.713 0.714 Ith (mA) 63 36 42 Jth (kA/cm2) 47 27 31 ฮทd (%) 38.6 60.8 61.9 highest ฮทwp (%) 23.5 34.8 28.3 Current required for highest ฮทwp(mA) 494 270 210 Output power at highest ฮทwp (mW) 100 72 55 0246810120 20 40 60 80 100 120Optical Power (mW)I(mA)1. w/o stressors3. side+topstressors2. sidestressors 77  (a) (b)  Figure 5-17 Current density field of structure 2 (a) in x direcdtion; (b) in y direction at 60 mA inject level. (a) (b)  Figure 5-18 Current density field of structure 3 (a) in x direcdtion; (b) in y direction at 60 mA inject level.   78 5.5 Effect of defect-limited minority carrier lifetime on the performance  For the study above, the defect-limited minority carrier lifetime ๐œ๐‘,๐‘› is set as 1 ns for conservative prediction. From Figure 5-19, we can see that by increasing the ๐œ๐‘,๐‘›, which means improving the material quality, the performance of laser improves greatly. When we have a better material quality, the carrier loss due to the SRH recombination is reduced, and as a result, the Ith decreases and ฮทwp increases. When the ๐œ๐‘,๐‘›  increase beyond 10ns, the reduction of SRH recombination rate has reached its limit and Auger recombination becomes the dominant carrier loss mechanism. Therefore, the decrease of Ith and the increase of ฮทwp reach their limit.  Technically, it is feasible to obtain Ge layers with better quality and longer carrier lifetime by approaches like Ge growth on a GOI (Germanium On Insulator) substrate [77] or direct wafer bonding and chemical mechanical polishing (CMP) [78]. Carrier lifetimes of  5.3 and 3.12 ns have been achieved respectively by the above approaches [77, 78] and a 10 ns  ๐œ๐‘,๐‘› is not too far away. Figure 5-20 shows the performance of the 3 structures with  ๐œ๐‘,๐‘› = 10 ๐‘›๐‘ . Ith decreases about 5-10 times and ฮทwp increases by 10% when ๐œ๐‘,๐‘› increases from 1 ns to 10 ns. .  Therefore, if a better material quality, along with the geometry and stress engineering, Ge laser performance will not be too far off from III-V laser performance in the range of Ith <50 mA and ฮทwp about 10-60%.   79 (a)0102030405060700 10 20 30 40 50Structure 1Structure 2Structure 3Threshold Current I th (mA)Defect limited carrier lifetime ๏ดn,p(ns) (b)202530354045500 10 20 30 40 50Structure 1Structure 2Structure 3Defect limited carrier lifetime ๏ดn,p(ns)Wall-Plug efficiency ๏จwp (%) Figure 5-19 Defect limited carrier lifetime dependence of Ith and ฮทwp. 80  Figure 5-20 . L-I curve for three structures with ๐›•๐ฉ,๐ง = ๐Ÿ๐ŸŽ ๐ง๐ฌ. Table 5-2    Laser performance of the three structures in Figure 5-20. Structure 1 2 3 ฮตeb (%) 0.25 0.713 0.714 Ith (mA) 12 4 5 Jth (kA/cm2) 8.8 3.0 3.7 ฮทd (%) 38.7 62.3 64.8 highest achievable ฮทwp (%) 27.0 43.8 41.1 Current required for highest ฮทwp (mA) 167 81 55 Output power at highest ฮทwp (mW) 35 24 17   0246810120 10 20 30 40 50 60Optical Power (mW)I(mA)1. w/o stressors2. sidestressors3. side+topstressors 81 Chapter 6: Thesis Summary and Suggestions for Future Work As an indispensable part in the optical interconnects, it cannot be denied that a silicon-compatible laser is the holy grail in the silicon photonics. Ge is the most compatible material with Si processing and has great potentials in the silicon-based electronics-photonic integrated circuits. Compared to III-V lasers on Si substrates, Ge-on-Si lasers can be processed in Si fabs with a much lower cost and much short development time to mass production. A major problem with Ge lasers is that the performance of initially demonstrated lasers were quite poor, and available stress engineering methods are not suitable for laser structure designs. In this work, two novel Ge laser structures with SiN stressors were proposed, modeled and optimized. SiN stressors were shown to be effective in reducing Ith and improving ฮทwp.  Side stressors turned out to be a more efficient way to increase ฮทwp than using the top and side stressors together. With the side stressors and geometry optimizations, a ฮทwp of 34.8% and an Ith of 36 mA (Jth of 27 kA/cm2) can be achieved with a defect limited carrier lifetime (๐œ๐‘,๐‘›) of 1 ns. With               ๐œ๐‘,๐‘› = 10 ๐‘›๐‘  , an Ith of 4 mA (Jth of 3 kA/cm2) and a ฮทwp of 43.8% can be achieved. These are tremendous improvements from the case without any stressors. These results give strong support to the Ge-on-Si laser technology and provide an effective way to improve the Ge laser performance.  However, to realize Ge lasers with reasonable performance, a lot more research efforts are needed both in crystal growth, processing, device design/fabrication and fundamental studies of Ge as an optical material. With a high tensile strain, Ge lasers can be used as infra-red lasers. Beyond making Ge lasers, for on-chip optical interconnects, many other aspects have to be addressed as well such as a photodiode suitable for receiving the wavelength from the Ge lasers and the integration scheme of optical devices and electronic devices on a wafer level. 82 Theoretically, a Ge photodiode with a higher tensile strain level has a smaller bandgap than a Ge laser with a lower tensile strain level and can serve as the desired photodetector. Again, stress engineering is the key.  Beyond Ge lasers, Ge can be used as an intermediate layer between InAs/GaAs lasers and Si to reduce crystal defects and thus lasing threshold. 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