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Impact of doping on epitaxial Ge thin film quality and Si-Ge interdiffusion Guangnan, Zhou 2017

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 Impact of Doping on Epitaxial Ge Thin Film Quality and Si-Ge Interdiffusion  by Guangnan Zhou   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  (Materials Engineering) THE UNIVERSITY OF BRITISH COLUMBIA  (Vancouver)  September, 2017   ©  Guangnan Zhou, 2017   ii  Abstract Germanium-on-silicon (Ge-on-Si) structure-based semiconductor devices are playing an increasingly important role in large-scale dense photonic integration, especially in silicon (Si) photonics. Si photonics has emerged as an effective solution to overcome the wiring limit imposed on integrated circuits with continued scaling by using optical interconnects instead metal interconnects. A Si-compatible laser is the last missing piece in optical interconnects on Si platforms. Recently, III-V lasers on Ge/Si substrates and Ge-on-Si lasers were demonstrated as the most promising candidates, in which Ge layers function as either the transition layers or the optical gain layers. For different applications, the requirements on Ge film quality and Ge/Si interface interdiffusion are different. Si-Ge interdiffusion during high-temperature growth or fabrication steps changes the distribution of Ge and increases atomic intermixing, which degrades device performance. However, studies on the doping impact on Ge film quality and Si-Ge interdiffusion are very limited, which were addressed in this work.  We investigate Ge-on-Si film quality systematically under different types of doping conditions (phosphorus, arsenic and boron) for the first time. It’s found that the boron doping significantly impairs the Ge film quality, while arsenic and phosphorus can effectively reduce the threading dislocation density without the commonly used defect annealing. This provides a new method to fabricate high-quality Ge-on-Si films, which can avoid undesired Si-Ge interdiffusion. Si-Ge interdiffusion with different doping at Ge/Si interfaces has been investigated experimentally and theoretically. The enhancement of interdiffusion was observed in n-type (phosphorus and arsenic) doped Ge-on-Si. The phenomenon is attributed to the Fermi-level iii  effect. A quantitative model of Si-Ge interdiffusion under high n-type doping was proposed. The model agrees well with the experimental data. This is also the first study on the quantitative modeling of Si-Ge interdiffusion with high n-type doping across the full Ge range. This work is of technical significance for the structure, doping and process design of Ge-on-Si structure-based devices including the two laser types mentioned above, Ge modulators and Ge photodetectors.   iv  Lay Summary Ge-on-Si structure based devices are widely used in computer and communication hardware. In this work, we studied ways of making Ge-on-Si of better quality. We were able to discover that the addition of certain elements, phosphorus (P) and arsenic (As), can enhance the Ge film quality. However, P and As also enhance the intermixing of Ge and Si at the Ge/Si interfaces during an annealing step, which is not desired. Another element, boron (B), turned out to make the Ge quality worse but doesn’t make the intermixing faster. This work mathematically modeled the impacts of P and As on the intermixing so that it can be predicted by calculations for structure and processing design.   v  Preface Most of the work in this thesis was done at the Department of Materials Engineering at UBC during the past two years. The author designed all the sample structures, doping conditions, and all the heating stage annealing conditions, performed heating stage annealing, prepared samples for secondary ion mass spectrometry (SIMS), etch pits density characterization, scanning electron microscope measurements, atomic force microscope measurements, X-ray diffraction characterization, all SIMS data analysis, modeling and simulations of Si-Ge interdiffusion.  Without the contributions from the collaborators, however, this thesis could not have been accomplished. The samples used in Chapter 3 were prepared by Dr. Kwang Hong Lee at Singapore-MIT Alliance for Research & Technology (SMART) from Prof. Chuan Seng Tan’s group. Transmission electron microscopy measurements and analysis in Chapter 3 were performed by Dr. Dalaver H. Anjum and Qian Zhang in King Abdullah University of Science and Technology supervised by Prof. Xixiang Zhang.    vi  Table of Contents  Abstract ....................................................................................................................................... ii Lay Summary ...............................................................................................................................iv Preface .........................................................................................................................................v Table of Contents ........................................................................................................................vi List of Tables................................................................................................................................ix List of Figures ...............................................................................................................................x List of Abbreviations .................................................................................................................xiii Acknowledgements................................................................................................................... xiv Dedication .................................................................................................................................. xv Chapter 1: Introduction, Literature Review and Problem Definition ..................................... 1 1.1 Industry Background and Motivations....................................................................... 1 1.1.1 Si photonics and Ge-on-Si applications ...................................................... 2 1.1.2 Ge-on-Si lasers: bandgap engineering and doping .................................... 5 1.1.3 III-V integration laser on Ge-on-Si substrates ............................................ 8 1.1.4 Requirements of Ge film quality, major Ge film growth methods, and quality comparison...................................................................................................... 10 1.2 Literature Review on Doping impact on Ge film Quality ......................................... 13 1.3 Literature Review on Si-Ge interdiffusion................................................................ 13 1.3.1 Si-Ge interdiffusivity in undoped relaxed crystals with low dislocation density ……………………………………………………………………………………………………………..14 1.3.2 Impact of strain and dislocations on Si-Ge interdiffusivity ...................... 16 1.3.3 Impact of dopants on Si-Ge interdiffusivity ............................................. 19 vii  1.4 Problem Definition and Thesis Goal ........................................................................ 20 Chapter 2: Literature Review of Si-Ge Interdiffusion Theories, Material Characterization, and Modeling Techniques ................................................................................................................ 22 2.1 Mechanisms and Models of Diffusion in Si-Ge Material System ............................. 22 2.1.1 Point defects: vacancy and interstitial ..................................................... 23 2.1.2 Doping introduced Fermi-level effect and modeling ............................... 25 2.2 Material Characterization Techniques in Si-Ge Interdiffusion Study ...................... 30 2.2.1 Secondary Ion Mass Spectrometry (SIMS) ............................................... 30 2.2.2 High-Resolution X-ray Diffraction (HRXRD) .............................................. 32 2.2.3 Measurement techniques for threading dislocation density................... 34 2.2.4 Atomic Force Microscope (AFM) .............................................................. 36 2.3 Interdiffusivity Extraction Method........................................................................... 37 2.3.1 The Boltzmann-Matano analysis .............................................................. 38 2.3.2 Time-averaged interdiffusivity –Boltzmann-Matano analysis ................. 40 2.4 Data Fitting and Numerical Simulations .................................................................. 41 2.5 Introduction to Kirkendall effect.............................................................................. 42 Chapter 3: Impact of Doping on Ge Film Quality .................................................................. 45 3.1 Structure Design, Growth, and Annealing ............................................................... 46 3.2 Etch-Pits-Density Results ......................................................................................... 50 3.2.1 EPD preparation and set up ..................................................................... 51 3.2.2 EPD results and discussions...................................................................... 53 3.3 Scanning Electron Microscope Results .................................................................... 60 3.4 Transmission Electron Microscope Results ............................................................. 61 3.5 Atomic Force Microscope (AFM) Measurements .................................................... 65 3.6 High-Resolution X-ray Diffraction (HRXRD) Results ................................................. 69 3.6.1 Kinematical theory underlying XRD.......................................................... 69 viii  3.6.2 HRXRD results and discussion .................................................................. 71 3.7 Analysis and Chapter Summary ............................................................................... 74 Chapter 4: Impact of Doping on Si-Ge Interdiffusion ........................................................... 76 4.1 Experiments ............................................................................................................. 76 4.1.1 SIMS results and discussions .................................................................... 76 4.1.2 The impact of different annealing procedures ........................................ 81 4.2 Quantitative Modeling of Si-Ge Interdiffusion with Doping.................................... 82 4.2.1 Extraction of Si-Ge interdiffusivity ........................................................... 83 4.2.2 Interdiffusivity extraction results and discussions ................................... 86 4.2.3 Effects from SIMS data smoothing and extraction .................................. 88 4.2.4 Annealing temperature extraction........................................................... 90 4.2.5 Mechanisms of interdiffusion enhancement ........................................... 94 4.2.6 Simulation of Ge profiles after annealing ................................................ 99 4.3 Chapter Summary .................................................................................................. 103 Chapter 5: Summary, Contributions and Future Work ....................................................... 104 5.1 Thesis Summary ..................................................................................................... 104 5.2 Contributions and Technological Implication ........................................................ 105 5.3 Future Work ........................................................................................................... 106 References............................................................................................................................... 109 Appendices .............................................................................................................................. 120    ix  List of Tables Table 1.1 Summary of the Ge film functions and requirements of different devices with Ge-on-Si structures. ................................................................................................................................. 12 Table 3.1 The 12 samples names and their corresponding doping and process conditions. ... 48 Table 3.2 Wafer offcut and Ge layer thickness information in all samples.. ............................ 50 Table 3.3 Time required to etch the whole Ge layer for each sample and the etching time chosen to observe and calculate the EPD.. ................................................................................... 52 Table 3.4 TDD value of the 12 samples measured by EPD.. ..................................................... 53 Table 3.5 TDD value of the 12 samples measured by EPD and SEM. ....................................... 61 Table 3.6 TDD values of the 12 samples measured by EPD, SEM (bold) and PVTEM (Italics). . 63 Table 3.7 The average and RMS surface roughness of samples after the second order of plane fitting. ............................................................................................................................................ 68 Table 3.8 Summary of the Ge films quality measured by AFM, HRXRD, EPD, SEM and TEM. . 75 Table 4.1 Intrinsic diffusion coefficients of different dopants (As, B and P) in Si/Ge at T = 850 °C [163-165]. ...................................................................................................................................... 80    x   List of Figures  Figure 1.1 Schematic of an all Integrated Electronic-Photonic Circuit. Figure courtesy of R. E. Camacho-Aguilera Ref. [7]. ........................................................................................................ 2 Figure 1.2 Schematic cross-section views of optoelectronic devices with Ge-on-Si structure. ..... 4 Figure 1.3 Schematic diagrams of band structures of Ge under different conditions. .................. 6 Figure 1.4 Summary of literature on the Si-Ge interdiffusivity data showing that the interdiffusivity has a strong dependence on Ge fraction. (T=900 °C) [91-96]. ........................ 16 Figure 1.5 Comparison between Si-Ge interdiffusivity in Dong et al. model (low TDD) and Gavelle et al. model (high TDD).T = 900 oC [94][101]. .......................................................................... 19 Figure 2.1 Schematic of vacancy-assisted diffusion mechanism. ................................................. 24 Figure 2.2 Schematic of (a) interstitial assisted kick-out diffusion; and (b) interstitialcy-assisted diffusion mechanism. ............................................................................................................... 25 Figure 2.3 Approximate location of charged vacancy energy levels in the silicon bandgap. ....... 26 Figure 2.4 A schematic structure of a SiGe epitaxial thin film with a MD and TD in it [116]. ...... 30 Figure 2.5 Schematic of SIMS process. ......................................................................................... 31 Figure 2.6 Illustration of X-ray diffraction process. ...................................................................... 33 Figure 2.7 A diffusion couple with an initial step profile for Boltzmann-Matano analysis. ......... 39 Figure 2.8 Illustration of Kirkendall et al.’s experiment................................................................ 43 Figure 3.1 Schematic diagrams of the structures in this work (a) B doped sample with high/low concentration; (b) As doped sample; (c) P doped sample; and  (d) undoped sample. ........... 46 Figure 3.2 Illustration of the two annealing temperature conditions (a) thermal cycles (5TC); and (b) merged high-temperature annealing (HT). ........................................................................ 48 Figure 3.3 Examples of Raman spectrum results of the same sample etched by the I2–based solution for different times (a) 15s; (b) 20s; and (c) 25s. The sample is A-No TC. .................. 52 Figure 3.4 Example of etching pictures from undoped samples. ................................................. 56 Figure 3.5 Example of etching pictures from phosphorus doped samples .................................. 57 xi  Figure 3.6 Example of etching pictures from arsenic doped samples .......................................... 58 Figure 3.7 Example of etching pictures from boron doped samples............................................ 59 Figure 3.8  Two examples of SEM measurement results, which are from sample (a)LB-HT and (b)LB-5TC.. ................................................................................................................................ 61 Figure 3.9 Images of PVTEM show different shapes and densities of threading dislocations in different samples ..................................................................................................................... 63 Figure 3.10 Cross section TEM images in bright mode of Sample (a) LB-No TC and; (b) LB-5TC.. 65 Figure 3.11 Examples of AFM measurement results from Sample U-No TC, P-No TC and P-5TC.66 Figure 3.12 HRXRD results of the samples (a) without annealing; and (b) after annealing.. ....... 73 Figure 3.13 HRXRD result of (2 2 4) reciprocal space mapping of Sample (a) U-No TC; and (b) P-No TC. ............................................................................................................................................. 74 Figure 4.1 Ge and dopants (As/P/B) profiles of samples without annealing measured by SIMS.. .................................................................................................................................................. 78 Figure 4.2 Ge profiles of samples with and without annealing measured by SIMS. .................... 79 Figure 4.3 dopant (As/P/B) profiles of samples with and without annealing measured by SIMS. .................................................................................................................................................. 79 Figure 4.4 Ge and dopants profiles in samples without annealing (w/o annealing), with 5TC and HT  (a) As doped samples; and (b) low concentration B doped samples. ............................... 81 Figure 4.5 Comparison between the extracted time-average interdiffusivities of different samples annealed under different thermal conditions.. ....................................................................... 82 Figure 4.6 Ge and dopant (As/P) profiles of the samples after annealing and without annealing. .................................................................................................................................................. 84 Figure 4.7 Ge and P profiles of P doped samples with 50 minutes annealing and 200 minutes annealing respectively. The annealing T = 870 oC.................................................................... 85 Figure 4.8 The time-averaged interdiffusivity as a function of Ge fraction using the Boltzmann-Matano method extracted from Sample U/P/A/HB/LB........................................................... 87 Figure 4.9 The extracted time-average interdiffusivity using the Boltzmann-Matano method with different smoothing methods .................................................................................................. 90 xii  Figure 4.10 Time-averaged interdiffusivity from Sample U (undoped) in comparison with the interdiffusivity calculations from literature models in [101] [94]; .......................................... 91 Figure 4.11 Comparison between the calculation results of the two intrinsic carrier density model at T = 890 oC. ............................................................................................................................ 98 Figure 4.12 Simulation results with different parameters (a) sample P-5TC; (b) sample A-5TC. ................................................................................................................................................ 102 Figure B1 A typical scenario of mismatch strain between two layers with a cubic crystalline structure……………………………………………………………………………………………………………………124 Figure B2 An identical multi-layered structure built in PANalytical Epitaxy simulation….……….126 Figure B3 Peak matching in PANalytic Epitaxy for sample U-No TC in Chapter 3 with R = 104% simulation………………………………………………………………………………………………………………… 126   xiii  List of Abbreviations 5TC Annealing between low temperature and high temperature 5 times (5×) A-5TC Arsenic doped sample with 5TC annealing A-HT Arsenic doped sample with HT annealing A-No TC Arsenic doped sample without annealing AFM Atomic force microscope EPD Etch pits density Ge Germanium HB-5TC High concentration B doped sample with 5TC annealing HB-No TC High concentration B doped sample without annealing HRXRD High resolution X-ray diffraction HT Annealing at high temperature (850 oC) for 50 minutes LB-5TC Low concentration B doped sample with 5TC annealing LB-HT Low concentration B doped sample with HT annealing LB-No TC Low concentration B doped sample without annealing 𝑛𝑖 Intrinsic carrier density P-No TC P doped sample without annealing P-5TC P doped sample with 5TC annealing PVTEM Plan-view TEM SEM Scanning electron microscope Si Silicon SIMS Secondary Ion Mass Spectrometry TD Threading dislocation TDD Threading dislocation density TEM Transmission electron microscopy U-5TC Undoped sample with 5TC annealing U-No TC Undoped sample without annealing 𝑥𝐺𝑒  Ge molar fraction XTEM Cross section TEM  xiv  Acknowledgements I would like to thank many people for their kind help and support during my master program in UBC. Firstly, I would like to express my deepest gratitude to my advisor, Prof. Guangrui Xia, for the continuous support of my master study and related research. Her guidance helped me in all the time of research and writing of this thesis. She has always been patient and generous to answer my questions. She actually revised my thesis word by word. I could not have imagined having a better advisor and mentor for my master study. I wish to express my sincere thanks to UBC Materials Engineering, for providing me with all the necessary facilities for the research.  I wish to give my special thanks to Prof. Chuan Seng Tan and Dr. Kwang Hong Lee at Singapore-MIT Alliance for Research & Technology (SMART). As the collaborators of this project, they provided us with the Ge-on-Si samples and important and insightful advice. This work would not be possible without their support. I am very grateful to Dr. Dalaver H. Anjum, Qian Zhang and Prof. Xixiang Zhang in King Abdullah University of Science and Technology. As the collaborators of this project, he helped us with the TEM characterization. I appreciate the help from my friends and colleagues at UBC. Special thanks are given to Dr. Mario Beaudoin, who trained me on cleanroom techniques and X-ray diffraction and gave me useful suggestions on my project. Jiaxin Ke, Weijun Luo, Tao Fang and Yunlong Zhao were/are my group-mates, and we had many interesting discussions during group meetings that greatly broadened my knowledge in microelectronics and photonics.  xv  Dedication    To my father and my mother.   1  Chapter 1: Introduction, Literature Review and Problem Definition 1.1  Industry Background and Motivations  Silicon (Si) based integrated circuits (ICs) have revolutionized the world ever since they were invented in the mid-20th-century. Driven by the huge demand, the semiconductor industry has improved the productivity of ICs by 25% ~ 30% annually [1]. According to the observations of Gordon Moore, the number of transistors per square centimeter on Si-based ICs doubles approximately every two years, also known as Moore’s law [2]. The most important and sophisticated transistors for digital ICs are metal-oxide-semiconductor field effect transistors (MOSFETs). However, this scaling trend has reached the physical limit of the thickness of gate dielectrics, where further reduction in the gate dielectrics thickness is not possible due to the unwanted high leakage current [3]. Meanwhile, the scaling of the transistors also leads to the increasing resistance-capacitance (RC) delay of metal interconnects [4]. Therefore, in the past decade, the semiconductor industry has slowed down the scaling pace greatly. For example, the scaling factor of the physical gate length from 2004 to 2010 is about 0.9 per year [5] compared to the predicted scaling factor of 0.7 [2].  The reason for that is that the metal wires connecting the transistors now become the dominant factor regarding the delay (shorter delay means faster speed) and power dissipation of ICs. The delay and power consumption of the metal interconnects are getting longer and higher when the technology node, the gate length of MOSFETs, decreases. The RC delay of metal wires increases drastically, whereas the intrinsic gate delay decreases slowly. The delay due to metal interconnects has become the limiting 2  factor of IC performance, known as the interconnect bottleneck. In analogy, the problem that we are facing for transistors and interconnects is like that of “fast cars on slow roads.” Because of this, broadband, high-density, high-speed data interconnections at a low-cost are strongly desired for signal processing and high-performance computing [6].   1.1.1 Si photonics and Ge-on-Si applications  Figure 1.1 Schematic of an all Integrated Electronic-Photonic Circuit. Figure courtesy of R. E. Camacho-Aguilera Ref. [7].  Optical interconnects via silicon (Si) photonic technology have been widely recognized as the most promising solution to overcome this bottleneck. The vision of the Integrated Electronic-Photonic Circuit is depicted in Figure 1.1. Si photonics refers to the field of using silicon as the platform for photonic systems. This area is gaining significant momentum because it allows for optical devices to be made cheaply using standard high volume and low-cost Si fabrication techniques and to be integrated with Si-based microelectronic ICs, also called “chips.” It provides a promising solution to cost-effectively meeting the ever-increasing demands on data speed and bandwidth [8]. Higher data rates 3  and higher interconnection densities with lower system-level power dissipations can be obtained using photons as the medium to detect, process, and transmit information [9]. Over the past two decades, an unprecedented technology boom has occurred in silicon photonics. Many key optical components with Ge-on-Si structures have been studied, such as waveguides [10], modulators [11][12] and photo-detectors [13][14][15] to transmit, modulate and detect light signals. They have been demonstrated with performance comparable to or even better than their III-V counterparts in certain aspects [16].  To transfer data through light signals in optical interconnects, a light source is a key component, which is also the most challenging device to achieve [17][18][19]. The development of an on-chip light source on Si has seriously lagged behind that of other photonic components mentioned above. As Si is an indirect band gap material, it has a low emission efficiency, which is usually one order of magnitude lower than the efficiency of III-V lasers. This has greatly limited the progress of optical interconnect technology. In the past few decades, researchers all over the world have invested extensive effort to find solutions to a Si-compatible lasing material system [16][ 20 - 23 ]. Among all of the candidates that have been extensively researched, Ge-on-Si lasers [24][25][26] and III-V quantum dot (QD) lasers grown on Ge-on-Si substrate have been demonstrated to be two of the most promising on-chip light sources [6]. The details of these two lasers will be elaborated in Section 1.1.2 and 1.1.3 respectively. The structures of these two lasers along with two other Ge-on-Si structure-based optoelectronic devices, Ge modulators, and Ge photodetectors, are shown in Figure 1.2.  4  In conclusion, Ge-on-Si structures are playing an increasingly important role in large-scale dense photonic integration because of the material compatibility and the maturing of key components in Si photonics [6][16]. In this work, we focus on its application in Ge-on-Si lasers and III-V QD lasers on Ge/Si substrates.      Figure 1.2 Schematic cross-section views of optoelectronic devices with Ge-on-Si structure; (a) a Ge optical modulator with Ge/Si1-xGex multi-quantum-wells (MQW) [27]. Reprinted with permission from © 2011, IEEE; (b) Ge/Si avalanche photodetector with p-i-n structure [28]. Reprinted with permission from © 2010, Nature Photonics; (c) Prototype of the electrically pumped Ge-on-Si laser device; Carrier recombination occurs in the n++ Ge region [29]. Reprinted with permission from © 2012, OSA; (d) an InAs/GaAs quantum dot (QD) laser diode on a Ge-on-Si substrate [30]. Reprinted with permission from © 2012, OSA.    5  1.1.2 Ge-on-Si lasers: bandgap engineering and doping  Ge is the most Si-compatible semiconductor because they have the same crystalline structure (diamond cubic structure), close lattice constants and similar physical properties such as self-diffusivity, Young’s modulus, Poisson ratio, etc. Ge plays an important role in Si photonics, such as in light sensing and modulation [31][32]. However, Ge is an indirect bandgap semiconductor as Si’s, which is inferior in light emitting applications than direct bandgap semiconductors, such as gallium arsenide (GaAs) and indium phosphorus (InP).  Figure 1.3 (a) illustrates the band structure of Ge [33]. Ge has the lowest energy point of its conduction band at the L valleys, and the energy difference between the L valley and the top of the valence bands is called the indirect bandgap, which is 0.66 eV at room temperature [34]. When an electron in an L valley recombines with a hole in the heavy hole (HH) or the light hole (LH) band, a third particle, normally a phonon, is needed to satisfy the momentum conservation principle, which makes the transition much more ineffective for light emitting compared to the transitions between the direct band and the valence band. Fortunately, the direct bandgap at the Γ valley is only 136 meV higher than the indirect bandgap at the L valley at room temperature. To turn Ge into an efficient light emitter, people have tried to compensate the difference between the direct and the indirect bandgap of Ge.  6  In 2007, researchers from MIT demonstrated that Ge could become a pseudo-direct bandgap material by adding tensile strain and/or high n-type (i.e. phosphorus/arsenic) doping [ 35 ]. The tensile strain can be introduced by thermal expansion coefficients mismatch or by stress concentration methods. As shown in Figure 1.3 (b), with an additional tensile strain applied to Ge, the Γ valley in Ge shrinks faster than the L valleys due to the smaller effective mass in the Γ valley. For Ge to become a direct band gap material, according to calculations and experiments, the applied tensile strain need to be at least 1.8% for biaxial strain or 4.6% for uniaxial strain  [33]. However, such high tensile strain narrows the bandgap too much so that the lasing wavelength will be larger than two microns [36]. Besides, it is technically hard to achieve such high tensile strain in Ge [37][38][39]. At the time of writing this thesis, there have been successful experimental efforts to change Ge into a direct bandgap material by introducing high uniaxial tensile strain up to 5.7% [36]. However, no successful efforts have been reported in making Ge-on-Si lasers with only tensile strain (no n-type doping).  Figure 1.3 Schematic diagrams of band structures of Ge under different conditions. (a) Ge without strain or doping; (b) tensile strained Ge without doping; (c) highly n-type doped Ge without strain.  7  To lower the requirement of tensile strain and keep the Ge lasing wavelength at around 1550 nm (within the telecommunication C-band wavelength range), researchers proposed to use n-type doping to raise the Fermi level and thus raise the effective bottom of the indirect L valleys to the Γ valley bottom or even higher [35]. As shown in Figure 1.3 (c), n-type dopants provide free electrons to occupy the energy states in the bottom of the L valleys. According to calculations and experiments, to turn Ge into a direct bandgap material, n-type doping level has to be on the order of 1019 cm-3 with 0.2% tensile strain Ge. Previously, many efforts have been made on adding large concentrations of n-type dopants into Ge. In 2009, Kurdi et. al. used gas immersion laser doping technique and achieved 5 × 1019 cm-3 of P doping concentration in Ge [40]. In 2012, the same doping level was achieved by Camacho et. al. with P delta-doped layers [41]. In addition, spin-on dopant process and multiple implantations were also successful in doping Ge up to about 1 × 1020  cm-3 [42][43].   By using n-type doping, in 2010, an optically pumped Ge-on-Si laser was demonstrated using 0.24% biaxial tensile strain [23]. It operated at room temperature with a gain of 50 cm−1 at an n-type doping level of 1 × 1019 cm−3. The lasing was in a wavelength range from 1590 to 1610 nm. In 2012, an electrically pumped Ge-on-Si laser was demonstrated by researchers from MIT and APIC Corporation with 4 × 1019 cm−3 n-type doped Ge and 0.2% biaxial tensile strain, as shown in Figure 1.2 (c). The lasing wavelengths were between 1520 nm and 1700 nm with a variation consistent with different clamping conditions and an output power up to 7 mW at room temperature [41]. 8   In the first demonstrated Ge laser, the threshold current density was too high (280 kA/cm2), and the quantum efficiency was too low for any practical use [41]. Our group recently performed two-dimensional laser simulations to investigate this problem [44]. The reasons for the poor performance are from three factors: 1) small optical confinement factor due to the laser structure design, which can be easily optimized, 2) small defect-limited carrier lifetime due to the material quality and processing imperfections and, 3) Si-Ge interdiffusion at the Ge/Si interface due to high P doping. Although the data of Si-Ge interdiffusion from the demonstrated Ge laser device is not available, according to the Ge laser simulations work of Li et al. [45] and Ke et al. [46] from our group, it is clear that Si-Ge interdiffusion is one of the key reasons for the poor performance. From the prototype structure of Ge laser in Figure 1.2 (c), the thickness of the Ge layer is on the order of 102 nm, which is thin enough to be susceptible to Si-Ge interdiffusion. Therefore, it is important to understand Si-Ge interdiffusion effect in the Ge-on-Si laser structures.  1.1.3 III-V integration laser on Ge-on-Si substrates  Another important application of Ge is to serve as the transition layer between III-V semiconductors (especially AlAs/GaAs) and Si substrates. One of the most obvious reasons is the very small lattice mismatch (0.08%) between GaAs and Ge. Direct bandgap III–V compounds have superb photonic properties for semiconductor emitters in a wide range of photonic applications [47][48], while Si is the most important material for main stream digital and analogy integrated circuits. 9  Therefore, the integration of III–V compound semiconductor materials, such as GaAs, on Si substrates have been extensively studied. In the long run, III-V lasers, monolithically grown on Si, are considered as the most promising on-chip lasers and may satisfy the requirements for low-cost, high-yield, temperature-insensitive and large-scale high-density monolithic integration [6].  One of the greatest difficulties in the monolithic integration of GaAs on Si substrates is the introduction of high-density threading dislocations (TDs) in the GaAs active region, due to the lattice mismatch between GaAs and Si [49][50]. The TDs are nucleated at the interface between the GaAs buffer layer and the Si substrate. Some of the TDs propagate into the III–V epitaxial layers grown on the buffer layer, leading to reduced optoelectronic conversion efficiency and lifetime of the device [49-51]. The TD density is sensitive to the nucleation layer in a material system, for example, much lower defect TD density is obtained within GaSb buffer layers grown on Si substrates with the use of an AlSb nucleation layer (NL) instead of a GaSb NL [52][53]. However, the application of the GaSb-based system is mainly in the field of mid-infrared wavelength devices, with their difficulties in material growth and device fabrication. Therefore, studies on developing new NLs to reduce the TD density within the GaAs buffer layer on Si substrates are critical for integrating GaAs-based photonic components with Si microelectronic circuits. Recent studies have shown the advantages in using Ge/Si substrates [54-57], due to the close lattice constant between GaAs and Ge. Therefore, the density of TDs in GaAs can be reduced significantly [58]. Moreover, high-quality Ge/Si substrates have become 10  commercially available [59-61].  III-V quantum dot (QD) lasers monolithically grown on Ge/Si substrates could thus be a better solution for the direct epitaxial growth of III-V lasers on Si substrates [30]. Utilizing Ge/Si substrates, Lee et at. described the first room-temperature (RT) continuous-wave operation of Si-based InAs/GaAs QD lasers, with low RT pulsed and continuous-wave threshold current densities [30].  In all the Ge-on-Si structure-based devices (Figure 1.2), different types of dopants can be involved. It’s crucial to understand their impact on the Ge/Si film quality for further application of these devices.   1.1.4 Requirements of Ge film quality, major Ge film growth methods, and quality comparison  The greatest challenge for high-quality Ge epitaxy on Si is the 4.2% lattice mismatch between the two materials. This mismatch causes two serious issues: high surface roughness resulting from the Stransky–Krastanov growth, and a high density of threading dislocations (TDs) in the Ge epitaxial layer. High surface roughness hinders the process of integrating Ge devices with Si electronics because complementary metal–oxide–semiconductor (CMOS) devices require planar processing, whereas a high density of TDs severely affects the performance of Ge photodiodes and lasers because of the recombination centers that are introduced along these dislocations [28]. For the Ge MOSFET, dislocations reduce carrier mobility and cause high leakage current. The first successful approach for growing high-quality epitaxial Ge layers on Si was 11  reported in a forward-looking paper by Luryi et al. in 1984 [62]. In this study, a graded SiGe buffer layer, grown in a molecular beam epitaxy chamber, was used to reduce the threading dislocation density (TDD) in the Ge layer. This method was later improved by Fitzgerald et al. using optimized SiGe graded buffer layers [63][64][65]. A low grading rate of ~10% Ge per micron was adopted to minimize dislocation nucleation rates. Choosing adequate growth temperatures for different SiGe compositions gives a high dislocation glide velocity but slow dislocation nucleation kinetics, allowing the film to relax mainly by gliding existing TDs instead of generating additional dislocations. At 50% Ge composition, an ex-situ chemical mechanical polishing step was used to remove crosshatch surface roughness and greatly reduce the dislocation pile-up formation that hinders dislocation gliding. For SiGe growth with Ge content of more than 50%, the growth temperature was gradually decreased to accommodate the reduction in the yield strength of the material and to prevent nucleation of the dislocations. This approach was used to successfully demonstrate Ge films with TDD less than 2 × 106 cm-2. High-quality epitaxial Ge-on-Si growth has also been achieved through careful direct Ge growth without using SiGe graded buffers layers. In this case, a two-step Ge growth technique is used to prevent islanding during the ultrahigh vacuum chemical vapor deposition (CVD) process, with subsequent annealing to significantly decrease the threading dislocation density [66][67][68]. In the initial growth step, a thin epitaxial Ge buffer layer of 30–100 nm is directly grown on Si at 320–360 oC. At such low growth temperatures, the low surface diffusivity of Ge kinetically suppresses the islanding of Ge. The growth temperature in the main growth step is increased to > 600 oC in order to 12  achieve higher growth rates and better crystal quality. If the low-temperature buffer layer is thick enough (> 30 nm), the Ge atoms are no longer influenced by the Ge and Si lattice mismatch and so are the homo-epitaxial Ge growth results. The films, therefore, remain flat even at elevated growth temperatures, and fully planar growth is achieved. The TDD of the as-grown (that is, without post-growth annealing) film is typical of the order of 107 – 109 cm–2. Adequate post-growth annealing at > 750 oC can reduce the TDD by up to two orders of magnitude to 106 cm-2 [69]. However, the thermal annealing process for reducing TDs is compromised by interdiffusion between Si and Ge layers, which is undesired for electronic devices. This two-step Ge-on-Si growth method was used to grow Ge-on-Si structures in this work.  Devices Ge on Si lasers InGa/GaAs lasers on Ge/Si Optical modulators Avalanche photodetectors Ge film functions Optical gain Transition layer SiGe quantum wells or Ge/Si Absorption layer Surface requirements Smooth Smooth Smooth Smooth Defect density requirements < 107 cm-2 < 107 cm-2 < 107 cm-2 < 107 cm-2 Strain requirements Tensile strain Relaxed Tensile strain Tensile strain Possible doping n++ type -- p+ type p+ type Si-Ge Interdiffusion Critical to minimize Not critical Critical to minimize Not critical Table 1.1 Summary of the Ge film functions and requirements of different devices with Ge-on-Si structures. 13   It is important to consider the Ge film functions and quality requirements for different optoelectronic devices with Ge-on-Si structures, which are summarized in Table 1.1. We can see that the requirements are different due to the different functions of Ge films.  1.2 Literature Review on Doping impact on Ge film Quality  In this work, we study the doping impact on the Ge film quality. Si-Ge interdiffusion is listed as a separate topic in the thesis title as it is a Si/Ge interface property, and not just about Ge films. Despite its increasingly important role in photonic integration, there is  a substantial lack of studies on the doping impact on Ge-on-Si film quality. Lee et al. studied the impact of high concentration arsenic (As) on Ge epitaxial film grown on Si (001) with 6o off-cut. He concluded that the TDD had been reduced by at least one order of magnitude to < 5 × 106/cm2 and attributed that to the enhancement in the velocity of the dislocation motion in an As-doped Ge film [70]. In the experimental work conducted at MIT [71], Si-Ge interdiffusion was observed. However, the Ge profiles were not calibrated, which were not suitable for any quantitative analysis.  1.3 Literature Review on Si-Ge interdiffusion Significant Si-Ge interdiffusion research efforts started in the 1990s. To measure interdiffusion at Si/SiGe interfaces, various techniques have been utilized by researchers, 14  such as Rutherford backscattering spectrometry [ 72 ], photoluminescence, X-ray diffraction (XRD) [73-76], secondary ion mass spectrometry (SIMS) [77-79] and Raman spectroscopy [80]. Typical interdiffusion structures studied in the 1990s were Si/Si1−xGex superlattices with thicknesses ranging from 30 nm to a few microns and with 𝑥𝐺𝑒 < 0.3. Due to the scaling of electronic and optoelectronic devices and the change in structures and fabrication techniques, the typical diffusion lengths of Si-Ge interdiffusion during thermal annealing in current technologies are comparable to the thickness of SiGe thin films in devices, which ranges from 1 to 102 nm. The 𝑥𝐺𝑒  range of current interest has expanded to the whole range of Ge. Therefore, since the last decade, Si-Ge interdiffusion has been revisited by many groups.  For Si-Ge interdiffusion in Ge-on-Si structures in electronic devices, Ge layers are typically under the conditions of (1) tensile strain due to thermal expansion mismatch, (2) high dislocation density, and (3) different types of doping concentrations. To summarize, previous research efforts on Si-Ge interdiffusion were discussed in the three aspects below: 1) Si-Ge interdiffusivity in  undoped relaxed crystals with low dislocation density, 2) The impact of strain and dislocations on Si-Ge interdiffusivity, 3) The impact of dopants on Si-Ge interdiffusivity.  1.3.1 Si-Ge interdiffusivity in undoped relaxed crystals with low dislocation density SiGe single crystals with low defect density are the commonly used for industry device applications. Point defects, i.e. interstitials (I) and vacancies (V), contribute to 15  dopant diffusion and interdiffusion in SiGe, which is now generally accepted based on both experimental observations and theoretical calculations. According to the literature, in Si and Si-rich SiGe alloys, Si and Ge self-diffusion are via both interstitials and vacancies [81-86]. Cowern et al. [78] demonstrated that for small values of Ge composition, diffusion at low temperature (T < 400 oC) is mostly mediated by vacancies. As the Ge fraction (𝑥𝐺𝑒) increases, the vacancy mechanism becomes more and more dominant [85-88]. Strohm et al. [86] concluded that when 𝑥𝐺𝑒  > 0.25, Ge self-diffusion is completely via vacancies. Moreover, with atomistic kinetic Monte Carlo simulations, Castrillo et al. [89][90] showed that both Si and Ge self-diffusion is controlled by vacancies over the medium Ge fraction range.  The interdiffusivity of Si-Ge has a strong concentration dependence. As summarized in Figure 1.3, the interdiffusivity increases almost exponentially with the Ge fraction[91-96]. Previously, utilizing XRD technology, people focused on the interdiffusion in multiple quantum wells (MQW) with 𝑥𝐺𝑒  < 0.50 [91][97] . In 2007, based on SIMS technology and the Boltzmann-Matano method, Xia et al. [93] built a DRDC model to describe the Si-Ge interdiffusivity with 𝑥𝐺𝑒  < 0.6. Recently, Dong et al. [94] established a benchmarking model for Si-Ge interdiffusivity over the full Ge fraction range. Based on Darken’s law and thermodynamics theory, Dong et al. considered Si-Ge alloys as regular solutions and expressed the Si-Ge interdiffusivity as follows: 𝐷 = 𝐷𝑆𝑖∗ (1 +𝜕𝑙𝑛𝛾𝑆𝑖𝜕𝑙𝑛𝑥𝑆𝑖) 𝑥𝐺𝑒 + 𝐷𝐺𝑒∗ (1 +𝜕𝑙𝑛𝛾𝐺𝑒𝜕𝑙𝑛𝑥𝐺𝑒) 𝑥𝑆𝑖,                                              (1.1) where 𝐷𝑆𝑖∗  and 𝐷𝐺𝑒∗  are the self-diffusivity of Si and Ge; 𝛾𝐺𝑒  (𝛾𝑆𝑖) stands for the chemical coefficient of component Ge (Si), which can be expressed as: 16  𝜕𝑙𝑛𝛾𝑆𝑖𝜕𝑙𝑛𝑥𝑆𝑖=(1−𝑥𝐺𝑒)𝑥𝐺𝑒(4017𝑥𝐺𝑒−17574)𝑅𝑔𝑇,                                                                        (1.2) 𝜕𝑙𝑛𝛾𝐺𝑒𝜕𝑙𝑛𝑥𝐺𝑒=(1−𝑥𝐺𝑒)𝑥𝐺𝑒(4017𝑥𝐺𝑒−17574)𝑅𝑔𝑇,                                                                       (1.3) where 𝑅𝑔 is the ideal gas constant and T is the absolute temperature.   The figure below compares the experimental data with Dong et al.’s benchmarking model over the full 𝑥𝐺𝑒, which agrees with literature data very well.  Figure 1.4 Summary of literature on the Si-Ge interdiffusivity data showing that the interdiffusivity has a strong dependence on Ge fraction. (T=900 °C) [91-96].  1.3.2 Impact of strain and dislocations on Si-Ge interdiffusivity Two decades ago, the first Si n- and p-MOSFETs with biaxial stress induced by Si1−xGex buffer layers were demonstrated by Welser et al. [98] and by Nayak et al. [99] respectively. A 2.2 times enhancement in electron mobility and a 1.5 times enhancement 17  in hole mobility were reported. Since then, strain has been widely applied in Si-based MOSFETs to enhance their performance.  It has been shown that compressive strain can significantly accelerate Si-Ge interdiffusion. Theiss et al. demonstrated that interdiffusion was enhanced in amorphous Si/Ge multilayers under hydrostatic pressure similar to that under lattice mismatch strain [100]. However, due to the strong concentration dependence of the activation energy of diffusion under pseudomorphic conditions, it was hard to separate the strain impact from the concentration impact. In 2002, using Ge self-diffusivity as the reference line, Aubertine et al. successfully separated strain and Ge concentration effects on Si-Ge interdiffusion [97]. Xia et al. built a DRDC model that obtained the Si-Ge interdiffusivity with the Boltzmann-Matano method under a relaxed condition and then modeled the impact of compressive strain as an exponential factor without any temperature dependence [93]. Recently, in 2014, Dong et al. built a compressive strain enhanced interdiffusion model via thermodynamics theory that extends the studied Ge molar fraction range from 0 < 𝑥𝐺𝑒 < 0.56  to 0 < 𝑥𝐺𝑒 < 0.75  [94]. Tensile strain, however, according to the observation by Xia et al. [93], has a negligible impact on Si-Ge interdiffusivity for the range of 0 < 𝑥𝐺𝑒 < 0.56. For real crystals, it is extremely hard to eliminate all of the dislocations no matter how perfect the crystals are. In the devices based on SiGe thin films, the most important extended defects are TDs and misfit dislocations (MDs). Threading dislocations can degrade the performance of electronic devices, which is undesirable. TDs can accelerate diffusion as rapid diffusion short circuits, which will be discussed in more detail in Section 18  2.1.2. However, the impact of TDs is dependent on their density and Ge fraction. In 2007, Xia et al. demonstrated that when the TDD is on the order of 107 cm-2, the impact of TDs on Si-Ge interfusion is only significant in the low Ge region (𝑥𝐺𝑒 < 0.05), compared to samples with TDD on the order of 105 cm-2. When the TDD is on the order of 1010 cm-2, reported by Gavelle et al. [101], for 𝑥𝐺𝑒 < 0.8, the interdiffusion is found to be strongly increased by the presence of dislocations. The comparison between the Gavelle et al. model (with high TDD ~ 1010 cm-2) and the Dong et al. model is illustrated in Figure 1.4. According to Gavelle et al. model, the interdiffusivity in a defect-free Ge-Si system has also been plotted and it’s consistent with Dong et al. Normally in electronic devices, TDs are minimized to the range of 104 ~ 106 cm-2 [102]. For Ge lasers with Ge-on-Si structure, however, TDs are often larger than 108 cm-2 at Ge/Si interfaces [103], which suggests that the impact of TDs cannot be neglected when considering Si-Ge interdiffusion in Ge-on-Si structure.  19   Figure 1.5 Comparison between Si-Ge interdiffusivity in Dong et al. model (low TDD) and Gavelle et al. model (high TDD).T = 900 oC [94][101].  1.3.3 Impact of dopants on Si-Ge interdiffusivity There have been a handful of studies on the doping impact on Si-Ge Interdiffusion. In 2002, Takeuchi and Ranade et al. [104][105] studied the Ge-Si interdiffusion in a polycrystalline Ge/Si structure under arsenic (As) doping. They reported that the interdiffusion was enhanced by about five times when the As doping level was 1 × 1021 cm−3. In 2008, Gavelle et. al. [101] studied the impact of boron on Si-Ge interdiffusion in a Ge on Si structure. The interdiffusion is retarded when the Ge layer is doped with boron. On the other hand, Ranade et. al. [105] reported that Si-Ge interdiffusion had been enhanced with boron doping in 2002. In 2011, Py et al. [106] studied the P impact on Si-Ge interdiffusion in a SiGe/Si superlattice structure. P enhanced interdiffusion has been 20  reported for SiGe alloys with a low Ge fraction (𝑥𝐺𝑒) of 0.265. He attributed the enhanced interdiffusivity to the indirect interaction between Ge and point defects released by P-defect clusters formed at high P concentration. Our group recently showed that high P doping greatly accelerates Si-Ge interdiffusion due to the Fermi-level effect [107][108]. Cai et al. [108] successfully established a quantitative model of interdiffusion with that theory. However, only one dopant (P) was involved and the Ge fraction was limited to 0.75 < 𝑥𝐺𝑒  < 1. The accuracy of this model requires more experimental data to fine-tune. Especially, interdiffusion with other dopants has not been well studied systematically, which was addressed in this work.  1.4 Problem Definition and Thesis Goal In conclusion, Ge-on-Si structures are playing an increasingly important role in large-scale dense photonic integration. However, Ge film quality and Si-Ge interdiffusion will severely affect performance of the electronic and photonic devices based on Ge-on-Si structures, including the threshold current, wall-plunge efficiency, and output power, etc. The impact from dopants on Ge film quality and Si-Ge interdiffusion still remain unclear. These limitations have introduced barriers for the use of Ge-on-Si substrates in the optoelectronic application, which were addressed in this work. In this thesis, Ge-on-Si structures with three different dopants (phosphorus, arsenic and boron) and without dopants were fabricated and annealed. Several different materials characterization methods have been performed to detect the film quality. The interdiffusion of Si and Ge of different films have been investigated experimentally and 21  theoretically.   This thesis mainly has two goals. One is to establish a quantitative model of Si-Ge interdiffusion under doping conditions across the full 𝑥𝐺𝑒  range. The other one is to investigate the impact of doping on Ge-on-Si films quality to find an approach to obtain Ge films with better quality for optical device applications. This subject is of technical significance for the structure, doping, and process design of Ge-on-Si based devices, especially for photonic applications.   22  Chapter 2: Literature Review of Si-Ge Interdiffusion Theories, Material Characterization, and Modeling Techniques After introducing the progress made in Ge-on-Si lasers and reviewing the prior-art on Ge film quality and Si-Ge interdiffusion, we need to discuss interdiffusion theories, experiment and modeling methods that are used in Chapter 3 and 4.   2.1 Mechanisms and Models of Diffusion in Si-Ge Material System Diffusion, on a macroscopic scale, is described by Fick’s laws for common cases. On a microscopic scale, however, it is a complicated many-body physics problem. When dealing with problems in this thesis, a constant diffusivity in Fick’s laws is no longer valid. Only by investigating the microscopic mechanisms can we explain or predict the macroscopic behavior of diffusion. In the semiconductor industry, the most widely used form of semiconductor materials is crystalline. For high-performance semiconductor devices, such as bipolar transistors, Ge modulators, Ge photodetectors, Ge lasers, and SiGe MOSFETs, 𝑆𝑖1−𝑥𝐺𝑒𝑥 (0 ≤ 𝑥 ≤ 1) materials are single crystalline. In SiGe crystals, defects play crucial roles in impurity diffusion and interdiffusion. Although there is a mechanism of diffusion that requires no defects, i.e., the direct exchange mechanism, where the diffusing atoms or ions simply exchange places with their neighboring host atoms/ions, it is usually regarded to be negligible due to the high activation energy needed [109].  23   Normally, in crystalline solids, defects can be categorized as point defects, line defects (dislocations), and area defects (grain boundaries and stack faults). In correspondence to these defects, there are three types of diffusion paths [110]: 1) Lattice diffusion mediated by point defects; 2) Dislocations-mediated short-circuit diffusion; and 3) Grain-boundary-mediated short-circuits diffusion. In this work, type 3) diffusion is not relevant as the SiGe samples in our studies are single crystalline. Previously, it was established that the diffusion in the SiGe system is mainly through lattice diffusion, i.e., point defects when the threading dislocation density (TDD) is on the order of 105 to 106 cm-2 [94]. Recently, according to Gavelle et al. and Dong et al.’s work [94][101], dislocations-mediated short-circuit diffusion is significant when the TDD is above 107 cm-2. For our samples, the TDDs are in the range of 108 to 1010 cm-2 in the region of interdiffusion, i.e., at the Ge/Si interfaces.  Therefore, in this thesis, we discuss only the first and second diffusion paths.  2.1.1 Point defects: vacancy and interstitial For Si (Ge) crystals, point defects, are defined as anything other than a Si (Ge) atom on a lattice site. There are two principal types of point defects that act as vehicles for diffusion. One is simply a missing Si (Ge) lattice atom or a vacancy, which we designate as V. The other is an extra Si (Ge) atom or an interstitial, which we designate as I. The corresponding diffusion processes with Vs and Is are illustrated in Figure 2.1 and 2-2.  24  The idea that diffusion in Si (Ge) happens through vacancy-assisted and interstitial-assisted mechanisms has been widely accepted based on both experimental observations and theoretical calculations [109]. As shown in Figure 2.1, a vacancy adjacent to a moving atom provides a mechanism for the moving atom to hop to the adjacent site. This is precisely the diffusion mechanism that dominates in  metals. However, in crystalline Si (Ge), a Si (Ge) interstitial can also “kick out” a substitutional atom from its lattice site, or simply diffuse as a bound pair (known as the interstitialcy mechanism) to diffuse along the bond direction, as illustrated in Figures 2-2 (a) and (b). In most literature, both the kick-out and interstitialcy diffusion processes are simply referred to as interstitial-assisted diffusion [111].  Figure 2.1 Schematic of the vacancy-assisted diffusion mechanism. [111]  25   Figure 2.2 Schematic of (a) interstitial assisted kick-out diffusion; and (b) interstitialcy-assisted diffusion mechanism. [111]  2.1.2 Doping introduced Fermi-level effect and modeling  Point defects in semiconductors can be neutral or exist in a numbered charged states. This is because point defects introduce energy levels into the band gap of a semiconductor [112]. Whether a defect is neutral or ionized depends on the position of the Fermi level. Consider the case with a high n-type doping level, as illustrated schematically in Figure 2.3 [111]. In this situation, EF is above the V- level. Thus, this level will be populated by an electron, i.e., the vacancy will be acting as an acceptor and will be charged negatively. It is therefore the dominant vacancy charge state will be V- and not the neutral vacancy.   26   Figure 2.3 Approximate location of charged vacancy energy levels in the silicon bandgap. Ei is the intrinsic (without doping) Fermi level. EF is the Fermi level in the N-type material. [111]  Let us consider a point defect 𝑋, and 𝑋𝜖(𝑉, 𝐼). Suppose that both Vs and Is exist in various charged states. The total concentration of defect X at thermal equilibrium can be written as [112]:  𝐶𝑋𝑒𝑞= 𝐶𝑋0𝑒𝑞+ 𝐶𝑋−𝑒𝑞+ 𝐶𝑋=𝑒𝑞+ 𝐶𝑋+𝑒𝑞+ 𝐶𝑋++𝑒𝑞 ,                                                                        (2.1) where the concentration of uncharged defects (𝐶𝑋0𝑒𝑞) depends on the temperature. The concentrations of charged defects can be presented as [80]:  𝐶𝑋−𝑒𝑞= 𝐶𝑋0𝑒𝑞𝑒𝑥𝑝 (𝐸𝐹 −𝐸𝑋−𝑘𝑇),                                                                                                  (2.2)  𝐶𝑋=𝑒𝑞= 𝐶𝑋0𝑒𝑞𝑒𝑥𝑝 (2𝐸𝐹 −𝐸𝑋−−𝐸𝑋=𝑘𝑇),                                                                                         (2.3)  𝐶𝑋+𝑒𝑞= 𝐶𝑋0𝑒𝑞𝑒𝑥𝑝 (𝐸𝑋+−𝐸𝐹𝑘𝑇),                                                                                                  (2.4)  𝐶𝑋++𝑒𝑞= 𝐶𝑋0𝑒𝑞𝑒𝑥𝑝 (𝐸𝑋+++𝐸𝑋+−2𝐸𝐹𝑘𝑇),                                                                                    (2.5) where 𝐸𝐹  is the Fermi level, which is a function of the doping concentration and the temperature:  𝑛𝑛𝑖= 𝑒𝑥𝑝 (𝐸𝐹 −𝐸𝑖𝑘𝑇),                                                                                                              (2.6) 27  Where 𝑛  is the electron concentration, and for high n-type doping level, 𝑛  ≈ dopant concentration; 𝑛𝑖  is the intrinsic electron concentration. Combining Equations (2.1) – (2.6), the total concentration of defects is given by:  𝐶𝑋𝑒𝑞= 𝐶𝑋0𝑒𝑞+ 𝐶𝑖,𝑋−𝑒𝑞 (𝑛𝑛𝑖) + 𝐶𝑖,𝑋=𝑒𝑞 (𝑛𝑛𝑖)2+ 𝐶𝑖,𝑋+𝑒𝑞 (𝑛𝑛𝑖)−1+ 𝐶𝑖,𝑋++𝑒𝑞 (𝑛𝑛𝑖)−2,                   (2.7) where 𝐶𝑖,𝑋𝑗±𝑒𝑞  denotes the equilibrium concentration under intrinsic condition (𝑛 = 𝑛𝑖)  for defect 𝑋  with charge state 𝑗 ± . As defects in various charge states act as diffusion-vehicles, diffusion in semiconductors is affected by the doping level. Similar to Equation (2.7), the diffusivity dependence on the electron concentration 𝑛  in an n-type doped semiconductor can be expressed as  [111]:  𝐷 = 𝐷0 + 𝐷− (𝑛𝑛𝑖) + 𝐷= (𝑛𝑛𝑖)2+ 𝐷+ (𝑛𝑛𝑖)−1+ 𝐷++ (𝑛𝑛𝑖)−2,                                         (2.8) where parameter 𝐷0, 𝐷− and 𝐷= are the diffusivity mediated by neutral, single negatively charged, and doubly negatively charged defects. Experimentally, for n-type doped cases, if the total interdiffusivity, 𝐷, is linearly dependent on the ratio 𝑛𝑛𝑖, the 𝐷− (𝑛𝑛𝑖) term is dominant, and the interdiffusion is mostly mediated by the single negatively charged point defects. If the total interdiffusivity, 𝐷, is quadratically dependent on the ratio 𝑛𝑛𝑖, the 𝐷= (𝑛𝑛𝑖)2 term is dominant, and interdiffusion is mostly mediated by double negatively charged point defects.  The diffusivity under intrinsic conditions, i.e., the doping level is less than 𝑛𝑖, is given by:  𝐷𝐴∗ = 𝐷0 + 𝐷− + 𝐷= + 𝐷+ + 𝐷++  ,                                                                            (2.9) 28  For atoms, such as P and As, as they provide extra free electrons, diffusion is mainly through neutral and negatively charged point defects. The fourth term, 𝐷+, and the fifth term, 𝐷++ are neglected in the literature [83]. With this simplification, for those that diffuse via neutral and negative charged point defects, the diffusion coefficient measured under extrinsic condition (𝑛 > 𝑛𝑖) can be elegantly described as [111]:  𝐷𝐴𝑒𝑓𝑓= 𝐷𝐴∗ (1+𝛽𝑛𝑛𝑖+𝛾(𝑛𝑛𝑖)21+𝛽+𝛾),                                                                                       ( 2 . 1 0 ) where 𝛽 = 𝐷−/𝐷0 and 𝛾 = 𝐷=/𝐷0. Expressed in this manner, the factor 𝛽 represents the linear variation in the diffusion coefficient in a doped material, which is related to 𝑋− point defects, and γ represents a square law variation in the diffusion coefficient in a doped material, which is related to 𝑋= point defects  [111].   Line Defects: Dislocations  It is well established that dislocations and grain boundaries act as rapid diffusion short circuits in metals and also in non-metals such as ionic materials and semiconductors [111]. In general, short-circuit diffusion proceeds with a lower activation energy than lattice diffusion.   In devices based on crystalline semiconductors, the two most common types of line defects are misfit dislocations (MDs) and threading dislocations (TDs). Since the lattice constant of SiGe is larger than Si, in the early stages of SiGe epitaxial growth, the in-plane lattice constant of SiGe is constrained by the Si substrate, resulting in the distortion of the cubic unit cell to a tetragonal cell. When an epitaxial layer is greater than 29  a certain critical thickness, it is energetically favorable to generate dislocations so as to relieve the strain in the epitaxial layer [113-115].  As is shown in Figure 2.4, these dislocations exist as dislocation half-loops in the epitaxial layer [116]. Each dislocation half-loop is combined with one misfit dislocation which lies in the planes parallel to the interface, and two TDs arms lie in the closely packed (111) planes which are inclined at 60o with respect to the interface.  MDs play a key role in strain relaxation in multilayered thin films while TDs can act as high diffusivity paths (the pipe effect) for diffusions in thin films [117]. As discussed in Section 1.3.2, the impact of TDs is related to the Ge fraction and their density. In this thesis, the TD density is an important parameter to evaluate the dislocations-related diffusion. The TD density is also important for subsequent III-V material growth on Ge and the performance of optoelectronic devices built on these III-V materials.  It should be noted that the TD density also will also change as the depth changes. For Si-Ge interdiffusion, it is at Ge/Si interfaces that the TD density is most relevant to Si-Ge interdiffusion. For TDs at the top of Ge surface, they are most relevant to III-V material quality and device performance.    30   Figure 2.4 A schematic structure of a SiGe epitaxial thin film with a MD and TD in it [116].    2.2 Material Characterization Techniques in Si-Ge Interdiffusion Study As mentioned in Chapter 1, besides temperature, four aspects will impact the Si-Ge interdiffusion, i.e. Ge concentration, strain status, TDs density, and doping concentrations. The first three aspects have been well studied. The impact of dopants (P, As and B) on Si-Ge interdiffusion is the topic of this study. Accordingly, in this work, different techniques were applied to characterize these impacting factors.    2.2.1 Secondary Ion Mass Spectrometry (SIMS)  The interdiffusion studied here is on the atomic scale, which requires high accuracy in characterizing concentration profiles. Secondary ion mass spectrometry (SIMS) is the most appropriate analytical technique for this study, because it has the highest detection sensitivity (~1016 cm-3 detection limit, which is 10-9 in molar fraction) for 31  measuring elemental concentrations, and can profile in the depth dimension with high precision (~Å scale) [111].   Figure 2.5 Schematic of SIMS process.    The principle underlying the SIMS technique is illustrated in Figure 2.5. A typical SIMS test area on a sample surface is around 200 × 200 µm  [111]. This area is physically sputtered by an incident ion beam, and the sputtered atoms from the sample are then collected from a smaller, central probe area. Among the sputtered atoms, those that are ionized can be accelerated by an electric field, mass analyzed and counted. By sputtering away the surface of the sample at a constant rate, a depth profile of the chemical species can be obtained.  SIMS is an excellent technique for profiling dopants in the SiGe system. However, a couple of effects in the SIMS analysis can influence the accuracy of the depth profile. First, in ultra-shallow depth profiling, a steady-state sputter rate may not be reached close to a surface or an interface, so the SIMS profile may be unreliable in regions near the Incident Ion BeamSputtered Atoms32  surface [111]. Second, there is a knock-on effect caused by the ion beam that physically strikes and recoils the near surface target atoms into deeper layers, thus degrading the depth resolution for the target atoms [118][119]. Third, profiling through a multilayer structure is difficult because the different materials have inherently different ion yields for the target atoms, leading to a “matrix effect” [111], which degrades the reliability of the measurement especially at the interfaces.    To achieve accurate depth profiles with minimal knock-on and mixing effect, a general solution is to combine a low ion beam energy with roughness suppression techniques [111]. In this thesis, samples in Chapter 3 and 4 were all performed SIMS measurements by Evans Analytical Group, which is the industry leader for commercial SIMS analysis with the best calibration for SiGe systems. The samples were sputtered with a 1 keV Cesium (Cs) ion beam which is obliquely incident on the samples at 60º off the sample surface normal. The sputter rate was calibrated by a stylus profilometer that measures the sputtered crater depth. With the known sputter rate variation with SiGe composition, the sputter rate was corrected on a point-by-point basis. The measurement uncertainty in Ge atomic fraction is ±1%. The depth/thickness uncertainty is about 5%.  2.2.2 High-Resolution X-ray Diffraction (HRXRD)  As strain impact diffusion greatly, it is important to monitor the strain status of the samples. HRXRD has been used for decades to investigate semiconductor structures. It is a powerful tool for the nondestructive ex-situ investigation of epitaxial layers. From HRXRD, information about the composition and strain can be obtained [111] [120]. 33   The principle of X-ray diffraction is illustrated in the figure below. The interaction of an X-ray beam with matter results in the scattering by the electrons of atomic constituents. The scattering process is most easily understood by considering the diffraction of the X-ray beam from a plane of atoms. From X-ray diffraction, the information about lattice constant can be obtained, which is determined by Bragg’s law: 2𝑑𝑠𝑖𝑛𝜃 = 𝑛𝜆 ,                                                                                                                 (2.11) where d is the spacing between diffraction planes, 𝜃 is the incident angle and 𝜆 is the wavelength of incident X-ray beam [121]. For a specified XRD equipment, 𝜆  is fixed. Strong diffraction occurs when incident angle 𝜃 satisfies Equation (2.11). Therefore, the measurement of the incident angle of the X-ray beam is essential for the accuracy of results. For modern HRXRD, angular precision of tens of arc-seconds is achieved [111].   Figure 2.6 Illustration of X-ray diffraction process.   34  The measurements of samples in Chapter 3 were performed in the Advanced Materials and Process Engineering Laboratory (AMPEL) at UBC using a PANalytical X’Pert PRO MRD with a triple axis configuration. The X-ray tubes were all operated at 45 kV and 40 mA in the line focus mode. The 𝐶𝑢 − 𝐾𝛼 wavelength (𝜆 = 1.5406 Å) was selected by a monochromator, giving a collimated monochromatic beam.   According to Equation (2.11), XRD measurements can obtain the lattice constant that is perpendicular to the characterized plane. However, the value is determined by both strain and chemical composition of the sample. In this work, the information of chemical composition was obtained from SIMS analysis. The calculation of strain in the epitaxial layer was then processed by PANalytical Epitaxy software to match the XRD peak separation between the peak from epitaxial layer and the peak from the substrate.  Details of the calculation using the software were attached in Appendix B.  2.2.3 Measurement techniques for threading dislocation density Conventionally, three methods are used to determine the dislocation density in semiconductor materials: plan-view transmission electron microscopy (PVTEM), cross-section transmission electron microscopy (XTEM) and etch-pit-density (EPD) observation. PVTEM observation is the most reliable method. However, since this method requires large magnification, the detection limit of the dislocation density is high, which is suitable to detect samples with high TDD ( > 108 cm-2) [122].  On the other hand, EPD observation is suitable to detect samples with low TDD ( < 106 cm-2). The process of EPD observation is usually done by etching and subsequent 35  counting the etch pits in an optical microscope. When an etching solution is applied on the surface of a piece of material, the etch rate at threading dislocations is faster than other regions due to the lower energy barrier, which resulting in pits on the surface. However, sufficient etching depth is required to form large pits that can be easily observed under a microscope. Therefore, when the etch pits are large, the value of TDD will be underestimated because multiple dislocations may overlap in one etch pit [123]. In terms of the measurement depth, EPD is not suitable to measure TD density in the top 200 nm. PVTEM sees defects at all depth and can’t differentiate the TD depth.  Overall, EPD observation is suitable when TDD is less than 106 cm-2, and PVTEM observation is suitable when the TDD is higher than 108 cm-2 [123][124]. In Chapter 3, both experiments have been applied to detect TDD in different samples. Meanwhile, XTEM has been conducted to observe the defect structure in the Si/Ge interlayer in typical samples.  In Chapter 3, samples with Ge-on-Si epitaxial structures were characterized by EPD measurement first. The Ge layers were etched with iodine (I2) solution for around 300 nm depth. The I2 solution is a mixture of CH3COOH (100 ml), HNO3 (40ml), HF (10 ml), I2 (30mg) [125]. The etch rate is approximately 40-80 nm/s depending on the different dopant configurations. After etching about half of the Ge layer, 4 – 6 different positions with a size of 160 μm × 120 μm on the surface were imaged by an optical microscope with bright/dark field modes, and the images were captured via a CCD camera. PVTEM and XTEM samples in Chapter 3 were prepared and imaged in the Advanced Nanofabrication, Imaging and Characterization Core Lab at King Abdullah University of Science and 36  Technology (KAUST) using a Titan3 80-300 TEM. The thickness of the TEM-specimens was estimated to be around 100 nm using low-loss EELS analysis. The TDs were estimated from these PVTEM images or XTEM images. To obtain the TDD from XTEM images, the number of TDs in a horizontal line were counted first. The dislocation density per unit length is the number of dislocations divided by the length. Finally, the TDD is the square of the dislocation density per unit length [126].   2.2.4 Atomic Force Microscope (AFM) AFM has been applied to characterize the surface roughness of the samples. It is a very high-resolution type of scanning probe microscopy (SPM), with demonstrated resolution of Å scale, more than 1000 times better than the optical microscope limit. It has been widely used as an effective surface morphology tool in the semiconductor area.  The measurements of samples in Chapter 4 were performed using Bruker Dimension Icon in the 4D LABS at Simon Fraser University. The AFM consists of a cantilever with a sharp tip (probe) at its end that is used to scan the specimen surface. The cantilever is silicon with a tip radius of curvature on the order of nanometers. When the tip was brought into proximity of a sample surface, forces between the tip and the sample lead to a deflection of the cantilever according to Hooke's law. Along with force, surface roughness was simultaneously measured.  The imagining mode was selected as “contact mode” in the experiments. In contact mode, the tip is "dragged" across the surface of a sample, and the contours of the surface are measured using the feedback signal required to keep the cantilever at a 37  constant position. Because the measurement of a static signal is prone to noise and drift, low stiffness cantilevers (i.e. cantilevers with a low spring constant, k) are used to achieve a large enough deflection signal while keeping the interaction force low. After the tip being brought into contact with the sample, it was raster-scanned along an x-y grid. The scanning size is 1μm × 1μm. Several different scanning areas in the samples have been chosen to calculate the surface roughness. Before the roughness calculations from the raw data, a plane fitting procedure must be performed since there are inevitably some tilt or bowing with the samples. The plane fitting procedure computes a single polynomial of a selectable order for an image and subtracts it from the image. However, too much plane fitting may lead to a smoother surface than reality. After the plane fitting, the roughness calculation was performed according to the heights of each pixel in the image [127]. Both the plane fitting and roughness calculation were conducted by the software Nanoscope Analysis 1.5 provided by Bruker Corporation.     2.3 Interdiffusivity Extraction Method In this work, Si-Ge interdiffusivities are extracted from the diffused Ge profiles of samples in Chapter 3 and 4 through the Boltzmann-Matano analysis. Details of this method are discussed below.     38  2.3.1 The Boltzmann-Matano analysis It is known that the equations governing common diffusion processes in solid materials are Fick’s laws. The first law and the second law are expressed as: 𝐽 = −𝐷∇C,                                                                                                            (2.12) 𝜕𝐶𝜕𝑡= ∇ ∙ (𝐷∇𝐶),                                                                                                   (2.13) where 𝐽 is the diffusion flux, D is the diffusivity/interdiffusivity, t is the time, and C is the concentration.  When the interdiffusivity is concentration dependent, such as in Si-Ge interdiffusivity, the shape of the diffused profile contains the information about the diffusivity. For example, a Gaussian profile is a result of constant 𝐷, while a box-like profile is a result of highly concentration dependent 𝐷. The interdiffusivity can be extracted by the Boltzmann-Matano analysis as a function of the concentration [128][129]. Based on Fick’s laws, the Boltzmann-Matano analysis is a graphical method to extract diffusivity from the shapes of diffused profiles. It has been widely used to extract interdiffusivity in binary metal alloys and SiGe alloys [130-133]. The analytical expression for interdiffusivity extraction by the Boltzmann-Matano analysis is given by: 𝐷(𝐶′) = −12𝑡(𝑑𝑧𝑑𝐶)𝐶=𝐶′∫ (𝑧 − 𝑧𝑀)𝑑𝐶𝐶′𝐶𝐿,                                                          (2.14) where 𝐷(𝐶′) is the interdiffusivity at Ge concentration 𝐶′, t is the annealing time, z is the depth, and 𝑧𝑀 is the position of the Matano plane. The definition for Matano plane 𝑧𝑀 should satisfy the following condition:   ∫ (𝑧 − 𝑧𝑀)𝐶𝑅𝐶𝐿𝑑𝐶 = 0,                                                                                           (2.15) 39  where 𝐶𝑅  and 𝐶𝐿  are two constant concentrations (R and L denote the right and left, respectively), as shown in Figure 2.7. In addition, for the analysis to be valid, it is required that the interdiffusivity be expressed as a function only of the local Ge fraction 𝑥𝐺𝑒 . Besides, the diffusion profile should satisfy the initial and boundary conditions:  𝐶(𝜂 = ∞) = 𝐶𝑅 ,                                                                                                           (2.16) 𝐶(𝜂 = −∞) = 𝐶𝐿,                                                                                                        (2.17) where η is defined as:  𝜂 =  𝑧√𝑡                                                                                                                            (2.18) It is worth mentioning that there are several sources of computational error that might occur in applying the Boltzmann-Matano method to experimental data derived from a diffusion couple: (1) uncertainty in the location of the Matano interface introduces error into the integral in Ea. (2.14); (2) the inverse gradient, (𝑑𝑧𝑑𝐶)𝐶=𝐶′, in Ea. (2.14)  Figure 2.7 A diffusion couple with an initial step profile for Boltzmann-Matano analysis. The green curve is the diffused profile.   CLCRDepth zConcentration CInitial Step Profilezm40  becomes extremely large far away from the Matano interface (i.e. dz/dC → ∞ as C → CR, and C → CL), which introduces large uncontrolled errors in the computed value of D(C) near the end compositions; and (3) the integral in Ea. (2.14) becomes so small at compositions lying near the ends of the diffusion zone that it, too, eventually introduces unacceptably large errors in computing the value of D(C) [134]. Data smoothing is often required for real data since it can remove noise and reduce these computational errors. One of the most common algorithms is the "moving average.” This method replaces each point in the signal with the average of "m" adjacent points, where "m" is a positive integer called the "smooth width." It is this method that we have used in Chapter 4 to smooth data. The algorithm was implemented by MATLABTM.   2.3.2 Time-averaged interdiffusivity –Boltzmann-Matano analysis In Xia’s analysis [60], when a thin and undoped SiGe epitaxial layer is grown on a Si substrate with a low dislocation density (<106 cm-2), the Si-Ge interdiffusivity will be a function of the Ge fraction only, which satisfies the condition of the Boltzmann-Matano analysis. However, for samples in Chapter 3 of this work, the condition that Si-Ge interdiffusivity is a function of the Ge fraction only could not be fully satisfied. Firstly, the TDD in our samples is high (≈108 cm-2) and is not constant throughout the whole depth. The distribution of TDs and its impact on the Si-Ge interdiffusivity are not dependent on 𝑥𝐺𝑒  only.  Second, the concentration of dopants in our samples are not constant, which directly influences the Si-Ge interdiffusivity. Moreover, during the annealing process, the 41  local dislocation densities and dopant concentration may change as a function of time, which makes the interdiffusivity time dependent.  Nevertheless, we can still extract an effective interdiffusivity through the Boltzmann-Matano analysis.  𝐷(𝐶′) = −12𝑡0(𝑑𝑧𝑑𝐶)𝐶=𝐶′∫ (𝑧 − 𝑧𝑀)𝑑𝐶𝐶′𝐶𝐿    (𝑇𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑇 = 𝑇0).        (2.19) In this case, the physical meaning of the effective interdiffusivity is: after annealing for time t0 at temperature T0, the diffused profile of the sample can be reproduced with the effective interdiffusivity, which is constant during the diffusion time.  In this work, before the analytical expression of the Si-Ge interdiffusivity is obtained, the time-averaged diffusivity was utilized as a reference, so that we could quantitatively compare the interdiffusivities of different samples with different thermal budgets.   2.4 Data Fitting and Numerical Simulations In this work, the interdiffusivity extractions and numerical simulations were performed with software MATLAB provided by Mathworks. MATLAB is a powerful tool for numerical computation and visualization. It allows for matrix manipulations, the plotting of functions and data, implementation of algorithms, etc.  In Chapter 4, we implemented our Si-Ge extrinsic interdiffusivity model with MATLAB to simulate the Si-Ge interdiffusion. The process of the interdiffusion was simulated by finite difference time domain (FDTD) method, which is a classical method 42  used for diffusion simulation [111].  Details of this method and the Matlab code is attached in Appendix A.   2.5 Introduction to Kirkendall effect The Kirkendall effect is a very important effect for doped Si-Ge interdiffusion. As Si diffuses much faster into Ge than Ge in Si, more atoms, and thus, more lattice sites are moved to the Ge side. The lattice sites originally on the Si side will reduce. The creation and annihilation of lattice sites might result in the shift of dopants profiles, which can, in turn, influence the Si-Ge interdiffusivity. The discovery of the Kirkendall effect, despite early controversies [135], initiated a serious paradigm shift in the history of solid-state diffusion theory. In 1947, Ernest Kirkendall and A.D. Smigelskas presented clear evidence that different atoms can have different diffusion coefficients in a system [136], strongly supporting the theory of diffusion by vacancy mechanism instead of the earlier accepted exchange or ring mechanisms [ 137 ]. In their paper, Kirkendall and Smigelskas presented results of experiments on interdiffusion of Zn and Cu in α-brass. The system was prepared by electroplating a brass bar with Cu and with Mo-wires on the interfaces to serve as inert markers. As illustrated in Figure 2.8, after heat treatments for different times and investigating the cross sections of the samples, they found that the markers move inwards parabolically with annealing time. The explanation for this phenomenon, now called Kirkendall-effect, is that the Zn atoms move faster outwards than the Cu atoms inwards. This causes the significant shrinkage of the inner brass, which cannot simply be explained 43  by the change of molar volumes alone. Darken made the first theoretical description using independent diffusion fluxes for the different constituents [138]. Later Seitz [139] and Bardeen [140] showed that diffusion by vacancy mechanism lead to Darken’s equations if local equilibrium of vacancy concentration is assumed. This means that for the Kirkendall effect to occur, on one side of the diffusion couple vacancies should be created, while on the other side they should annihilate. Other than the marker movement this can manifest in numerous other phenomena. These are formation of Kirkendall voids [141][142], the generation of internal stresses [142][143] and macroscopic deformation of the samples [144]. [145]   Figure 2.8 Illustration of Kirkendall et al.’s experiment. The inert markers (Mo) will move inwards as Zn atoms move faster outwards than the Cu atoms inwards. [136]    44  To our knowledge, there is currently no experimental study of Kirkendall effect in Si-Ge interdiffusion system due to the lack of inert marker in atomic scale. Nor was there any theoretical study.  In this work, we focus on the shift of dopants profiles during the annealing and the relation with Kirkendall effect.   45  Chapter 3: Impact of Doping on Ge Film Quality As introduced previously in Chapter 2, Ge-on-Si structures are important for Si-compatible laser applications. Whether we achieve Si compatible lasers with Ge-on-Si lasers or III-V lasers on Si with a Ge transition layer, there will be certain requirements for Ge film quality. Doping is commonly used in semiconductors to change the free carrier concentration and thus the conductivity. Ge has been widely used and well-studied, however, the use of doped Ge on doped Si has been sparse. It is likely due to the fact that the most mature Ge device type is likely a Ge photodiode with the doping configuration as p-type/intrinsic/n-type (p/i/n). Commonly Ge serves as the intrinsic region, and the bottom p-type region can be on Si and the top n-type parts can be polycrystalline Si (poly-Si) or metals respectively. Therefore, Ge has been mostly used without doping, and the impact of dopants on the Ge-on-Si film quality remains unclear. This chapter focuses on the impacts of three dopants (As/P/B) on that. Samples with different doping configurations have been designed and annealed. Five different materials characterization methods have been applied to characterize the samples quality.  This subject is of technical significance for the structure, doping, and process design of Si compatible lasers and other Ge-on-Si devices.  46  3.1 Structure Design, Growth, and Annealing  Figure 3.1 Schematic diagrams of the structures in this work (a) B doped sample with high/low concentration; (b) As doped sample; (c) P doped sample; and  (d) undoped sample.   12 Ge-on-Si samples with 5 doping configurations (Figure 3.1) and 3 annealing conditions (no annealing, 5 thermal cycles, and merged annealing) were designed with different doping and annealing configurations. Ge-on-Si layers form Ge/Si interdiffusion couples. Sample U, A, P, B stand for undoped Ge/Si, arsenic (As) doped Ge/Si, phosphorus (P) doped Ge/Si, and boron (B) doped Ge/Si respectively. Furthermore, we designed two different boron concentrations to study boron doping level and the Fermi level effect for B doped Ge/Si. One is with higher boron concentration (HB), and the other one is with a lower concentration (LB). The B concentration in Sample LB is lower than the intrinsic 47  carrier density of Ge (𝑛𝑖_𝐺𝑒), and dopants concentration in Sample P, A and HB are higher than that. P and As are the highest concentration currently we can grow in the epitaxial reactor.   All samples were grown on 8-inches (100) Czochrolski (CZ) Si wafers in a metal-organic chemical vapor deposition (MOCVD), Aixtron CRIUS CCS model by Dr. Kwang Hong Lee at the Singapore-MIT Alliance for Research & Technology (SMART) from Prof. Chuan Seng Tan’s group. Before a doped Si layer was deposited, the Si substrate was treated at 1050 ± 10 oC for 10 minutes under H2 ambient at 400 mbar. Then, a 600 nm doped or undoped Si layer was deposited at 950 ± 10 oC under H2 ambient at 100 mbar. To improve the Ge film quality and reduce the threading dislocation caused by the Ge-Si lattice difference, a 100 nm doped/undoped Ge seeding layer was deposited at 400 ± 10 oC under H2 ambient at 100 mbar (low-temperature Ge growth) on top of the Si layer. Finally, a doped or undoped bulk Ge (~600 nm) was deposited at 650 ± 10 oC under H2 ambient at 100 mbar (high-temperature Ge growth). All the dopants quoted were in-situ doped during the growth of the corresponding layers.  Immediately after the growth procedure, half of  the samples were annealed while another half were left unannealed (No TC) as a comparison. Post-deposition thermal cycling was performed by repeating a hydrogen annealing cycle between low annealing temperature (LT) and high annealing temperature (HT) ranging from 600 °C to 850 °C for 5 times (5×). The temperature profile is illustrated in Figure 3.2 (a). The temperature was higher than the growth temperature of the bulk Ge layer at 650 °C. Each annealing step at HT was 10 minutes, at LT was 5 minutes, and the annealing was performed in an H2 48  environment to improve the quality of the Ge epitaxial film.  The ramping up rate was around 1°C/s and the cooling down rate was also around 1 °C/s.   Figure 3.2 Illustration of the two annealing temperature conditions (a) thermal cycles (5TC); and (b) merged high-temperature annealing (HT).  Sample Name Doping and Process Conditions Sample Name Doping and Process Conditions U-No TC Undoped sample without annealing A-No TC As doped sample without annealing U-5TC Undoped sample with 5TC annealing A-5TC As doped sample with 5TC annealing P-No TC P doped sample without annealing A-HT As doped sample with HT annealing P-5TC P doped sample with 5TC annealing LB-No TC Low concentration B doped sample without annealing HB-No TC High concentration B doped sample without annealing LB-5TC Low concentration B doped sample with 5TC annealing HB-5TC High concentration B doped sample with 5TC annealing LB-HT Low concentration B doped sample with HT annealing Table 3.1 The 12 samples names and their corresponding doping and process conditions.   Besides the 850 oC for 10 minutes and 650 oC for 5 minutes for total 5 cycles (5TC), we also had two extra samples annealed at 850 oC for 50 minutes, which 49  was isothermal annealing, as seen in Figure 3.3 (b). This was designed to check the impact of the different temperature profiles with the same high T annealing time. This was done by ramping up the temperature immediately after the bulk-Ge layer growth. Then, it was held at 850 oC for 50 minutes. The two samples are arsenic doped sample (sample A-HT) and low concentration boron doped sample (LB-HT) respectively. They were designed to study the impact of different annealing procedure and potential influence of the low temperature (650 °C). The 12 samples and their corresponding annotations are listed in Table 3.1, and they will be used in the following text.  It is worth elaborating the temperature control and record in more details before moving on. The system came with two different temperature reading systems. One was a thermal couple below the susceptor to set the heating power, and another was a multi-channel pyrometer to read the wafer surface temperature which allowed real -time measurements. The MOCVD model is CRIUS CCS from Aixtron. Usually, the wafer surface temperature is about 100-150 oC lower than the thermal couple temperature. Especially, the surface temperature of the As doped sample was always about 20 oC slightly lower than the other specimens, even though they were targeted at the same thermal couple temperature setting. The 850 and 650 oC quoted so far were the surface temperature readings of samples other than the As-doped ones. The temperature values are further discussed in 4.2.4, where the temperature values are compared with other interdiffusion studies. The undoped, P and As doped wafers are 6° off-cut towards the [110] direction. The offcut was aimed at reducing the antiphase domains for the future study of the 50  materials properties of III-V materials that are grown on different Ge substrates. However, the B-doped Ge is too rough for the subsequent III-V materials growth, so Dr. Lee used the Si substrates to on-axis wafers for the B-doped Ge growth except for the sample LB-5TC as a control sample to compare with LB-5TC. The wafer orientation information is listed in Table 3.2. After the annealing procedure, the thickness of Ge layer was measured by an ellipsometer, the model of which is J. A. Woollam, RC2. The thickness information is also listed in Table 3.2. The thickness uniformity across the wafer is less than 1%. The thickness variations between wafers are due to run to run variations and doping impact. According to our experiments, For As-doped Ge, the growth rate is only about 10% of the undoped Ge. The growth rate of P-doped Ge is about 70% of the undoped Ge; whereas the growth rate of B-doped Ge is similar to the undoped Ge.  Ge sublimates during the 850 °C heating step, which is estimated to be 20-70 nm. Sample  Wafer offcut Ge thickness(nm)  Sample Wafer offcut Ge thickness(nm) U-No TC 6° 746.2  A-HT 6° 572 U-5TC 6° 808  LB-No TC 0° 629 P-No TC 6° 712  LB-5TC 6° 634 P-5TC 6° 696.4  LB-HT 0° 587 A-No TC 6° 611  HB-No TC 0° 666 A-5TC 6° 636  HB-5TC 0° 673  Table 3.2 Wafer offcut and Ge layer thickness information in all samples. The offcut is towards [110].  3.2 Etch-Pits-Density Results Etch-pits-density (EPD) methods have been widely applied to characterize the threading dislocation density (TDD) in semiconductors. The experiments will be 51  elaborated in detail in this section. All the experiments were performed in Advanced Materials and Process Engineering Laboratory (AMPEL) at UBC using a wetbench.  3.2.1 EPD preparation and set up The Ge layers were etched with iodine (I2)-based solution. The I2-based solution is a mixture of CH3COOH (100 ml), HNO3 (40ml), HF (10 ml), I2 (30mg) [125]. The etch rate is approximately 40-80 nm/s depending on the dopants and doping level.  In the experiment, it was important to etch to roughly half of the Ge layer. Since there were always high-density TDDs in the Ge seeding layer, etching too deep will overestimate the TDD results for Ge film quality. Etching too shallow is not enough to show the etch pits and thus not desirable. To measure the etching rate for different samples, we first etched the samples in different time ranging from 8s up to 40s. Then, we performed Raman spectrum measurement to get the elements concentration information left at the surface for different samples etched by different times. Thus, we can decide the time required to etch the whole Ge layer for each sample. An example of Raman spectrum results is illustrated in Figure 3.3. Figure 3.3 (a) shows the result of 15s etching, at which the remaining Ge was thicker than the Raman penetration depth. Figure 3.3 (b) indicates that more Ge has been etched and it’s thinner than the Raman penetration depth, while the Ge layer has been etched entirely in Figure 3.3 (c). Finally, the etching time we chose for observation and calculation of EPD for each sample are listed in Table 3.3.  52    Figure 3.3 Examples of Raman spectrum results of the same sample etched by the I2–based solution for different times (a) 15s; (b) 20s; and (c) 25s. The sample is A-No TC.  Sample code Etching time for the whole Ge layer (s) Etching time for observation (s) U-No TC 40 15,20 U-5TC 40 12,15,20 P-No TC 30 10,15,20 P-5TC 16 5,7,9 A-No TC 20 10,15 A-5TC <20 7,8,9 A-HT <20 7,8,9 LB-No TC 30 15 LB-5TC >20 12,14,15 LB-HT 15 8,10 HB-No TC 30 15 HB-5TC >20 12,15 Table 3.3 Time required to etch the whole Ge layer for each sample and the etching time chosen to observe and calculate the EPD. Some samples have been etched in different time.  53  After etching about half of the Ge layer, 4 – 6 different positions with a size of 160 μm × 120 μm on the surface were imaged by an optical microscope with bright/dark field modes, and the images were captured via a CCD camera. Several different etching times for the same sample were selected to check EPD’s depth dependence. When the etch depth hasn’t reached the Ge seeding layer, no significant difference has been found when etching time was different.  3.2.2 EPD results and discussions Table 3.4 lists the threading dislocation density (TDD) values in the Ge films of the 12 samples measured by EPD. The details of EPD pictures are listed from Figure 3.4 to Figure 3.7. The TDDs in some samples are too high for accurate EPD measurement. When the etching pit density is too high, etch pits can overlay making accurate counting not possible, in those cases, we recorded the TDD as > 2 × 108 cm-2. Sample  TDD value (cm-2) Sample  TDD value  (cm-2) Sample  TDD value (cm-2) U-No TC 3.5 ± 1.5 × 106 A-No TC 5 ± 3 × 105 LB-5TC (6o) 1.2 ± 0.5 × 108 U-5TC < 5 × 104 A-5TC 1.2 ± 0.5 ×105 LB-HT (0o) 1.1 ± 0.5 × 108 P-No TC 3 ± 1 × 105 A-HT 1.2 ± 0.5 × 105 HB-No TC (0o) > 2 × 108 P-5TC 1.75 ± 1 × 105 LB-No TC (0o) > 2 × 108 HB-5TC (0o) > 2 × 108  Table 3.4 TDD value of the 12 samples measured by EPD. The boron doped samples have either 0° or 6° offcut towards [110] direction and are labeled. All other samples are 6° offcut towards [110] direction.  54  Comparing TDD values of the samples after annealing with different dopants configurations, we can easily found that the boron doped samples have the highest values (> 108 cm-2), which means they have the poorest quality. A Ge layer with such dense dislocation is not suitable as a transition layer between GaAs and Si, as the subsequent GaAs growth will be affected. The merged annealing doesn't have much impact on the TDD value compared with the corresponding 5TC annealing.    For As and P doped samples, for samples without annealing, TDDs in sample A-No TC and P-No TC are in the 105 to 106 cm-2 range, one order of magnitude lower than that in U-No TC. This TDD level is already low enough for electronic or photonic applications, and there is no need to have an extra defect annealing step for A-No TC and P-No TC. This also prevents the interdiffusion during the defect annealing step and is one of the major findings of this work. On the other hand, TDDs in A-5TC and P-5TC are higher than that in U-5TC. This can be interpreted by the following pictures: (1) As previously reported, dislocations move faster in As-doped Ge and slower in Ga-doped than in undoped Ge [146]. This is due to the presence of shallow donor or acceptor levels at the dislocation or other defects such as kinks or antiphase defects. A similar explanation can be applied to P/B. We expect a suppression of dislocation generation in As/P doped Ge and prompting in B doped Ge. Thus, P/A-No TC has a lower TDD value, and LB/HB-No TC has a higher value than U-No TC. (2) In Ge/Si, it is accepted that the dislocation core is a perfect sink for impurity atoms that arrive there [147][148].  Impurities are known to be effectively gettered by dislocations [149]. Dislocations can be 55  immobilized due to the formation of impurity complexes or clusters at dislocation si tes through their accumulation [150]. Thus, P/A-5TC has a higher TDD than U-5TC. In conclusion, the boron doping significantly impairs the Ge film quality, while As and P can reduce the TDD level in unannealed samples. This provides a new method to fabricate high-quality Ge-on-Si films without defect annealing procedure, which can avoid undesired Si-Ge interdiffusion.   56                     Figure 3.4 Example of etching pictures from undoped samples; (a) U-No TC no etching; (b) U-No TC 15s etching; (c) U-No TC 20s etching; (d) U-5TC No etching; (e) U-5TC 12s etching and; (f) U-5TC 15s etching. The TDD value has been reduced by around two orders of magnitude due to the 5TC annealing procedure. Etching pits are pointed out by white arrows in Figure 3.4 (f).   Some pits, like those in Figure 3.4(b), might be difficult to be observed in printed version.   57                  Figure 3.5 Example of etching pictures from phosphorus doped samples; (a) P-No TC no etching; (b) P-No TC 10s etching; (c) P-No TC 15s etching; (d) P-5TC No etching; (e) P-5TC 9s etching and; (f) P-5TC 10s etching. P-No TC has a lower TDD value than U-No TC. However, the 5TC annealing procedure hasn’t reduced TDD as much as undoped samples.   58         Figure 3.6 Example of etching pictures from arsenic doped samples; (a) A-No TC no etching; (b) A-No TC etching for 15s; (c) A-No TC etching for 10s; (d) A-5TC No etching; (e)A-5TC etching for 7s and; (f) A-HT etching for 9s. The As doped samples have similar TDD results with P doped samples.   Compared with other samples, the A-5TC has a clear cross-hatch pattern, which is very common in Ge-on-Si films.    59                                                                                                     Figure 3.7 Example of etching pictures from boron doped samples; (a) LB-No TC no etching; (b) LB-No TC 15s etching; (c) HB-No TC 15s etching; (d) LB-5TC 12s etching; (e) LB-5TC 14s etching; (f) LB-HT 15s etching; (g) LB-HT 10s etching and; (h)HB-5TC 15s etching. 60  3.3 Scanning Electron Microscope Results As illustrated in Figure 3.7, the pits in the boron doped sample are so dense that we could hardly count those pits due to the magnification limitation.  Also, it is difficult to distinguish the bumps from the pits in some samples, since they look rather similar under an optical microscope. A much higher degree of magnification is required To address problems. Therefore, scanning electron microscope (SEM) was used to measure the boron doped samples after etching. The measurement was performed by Hitachi S3000N VP-SEM with EDX, at the Materials Electron Microscopy Lab at UBC. Figure 3.8 has two examples of SEM measurement results, which are from sample LB-5TC with 12s etching and LB-HT with 9s etching. Those small dots we got from EPD optical microscope images were confirmed to be etch pits caused by threading dislocations. The TDD values in these samples measured by SEM are estimated to be around 1.5 × 108 cm-2, which is a little higher than what we got from EPD measurement. That inconsistency is quite reasonable and expected as SEM is much better in resolving overlapping pits and pits that are too shallow or too small to be observed by optical microscope imaging. All these results in an underestimation of the TDD value measured by EPD. Table 3.5 summarizes the TDD measurement results by EPD and SEM. 61        Figure 3.8  Two examples of SEM measurement results, which are from sample (a)LB-HT and (b)LB-5TC. The scanning area is estimated to be 12.72 μm × 9.54 μm. The TDD values are estimated to be 1.5 × 108 and 1.6 × 108 cm-2 respectively.   Sample  TDD value (cm-2) Sample  TDD value  (cm-2) Sample  TDD value (cm-2) U-No TC 3.5 ± 1.5 × 106 A-No TC 5 ± 3 × 105 LB-5TC (6°) 1.5 × 108 U-5TC < 5 × 104 A-5TC 1.2 ± 0.5 ×105 LB-HT (0°) 1.6 × 108 P-No TC 3 ± 1 × 105 A-HT 1.2 ± 0.5 × 105 HB-No TC (0°) > 2 × 108 P-5TC 1.75 ± 1 × 105 LB-No TC (0°) > 2 × 108 HB-5TC (0°) > 2 × 108  Table 3.5 TDD value of the 12 samples measured by EPD and SEM. The boron doped samples have either 0° or 6° offcut towards [110] direction, and are labeled. All other samples are 6° offcut towards [110] direction. The bold fonts are from SEM measurements, and all others are from EPD measurements.  3.4 Transmission Electron Microscope Results Plan View TEM (PVTEM) and cross section TEM (XTEM) have been performed by Dr. Davaler Anjum and Qian Zhang at Imaging and Characterization Lab, King Abdullah University of Science & Technology (KAUST), Saudi Arabia. These were used to detect the total TDD, TDD in the Ge bulk layer and Ge seeding layer respectively. 62  Figure 3.9 shows PVTEM images of the Sample LB-No TC and LB-5TC. Dislocations along different directions will demonstrate distinctive shape in PVTEM. Misfit dislocations (MDs) (Burger vectors <110>) will form lines in the images as shown in Figure 3.9 (a). Meanwhile, TDs will show as pits, some of which are pointed out by white arrows in Figure 3.10 (a).  Comparing the samples of the same dopant configuration before and after annealing, we can easily find that the annealing procedure can significantly reduce the TDD in the Ge bulk layer. That conclusion is consistent with what we found by EPD in Section 3.2.2. Besides, there are much more TDs in the boron doped samples compared with undoped samples, which is also in accordance with the previous conclusion. On the other hand, we could hardly detect any pit in sample U-No TC, the dislocations are mainly MDs. According to the EPD experiments, the TDD value of U-No TC is 2e6~5e6 cm-2. The image size in Figure 3.9 (a) is 3 μm × 3 μm, which means we need 5 such pictures to get one pit on average. It is not surprising that there is no pit in one picture, let alone the sample U-5TC with an even lower TDD value.     (b) 63   Figure 3.9 Images of PVTEM show different shapes and densities of threading dislocations in different samples; (a) LB-No TC and; (b) LB-5TC.  In conclusion, PVTEM has been performed to detect the TDD values in the Ge bulk layers of different samples. The results are consistent with EPD results. Furthermore, it is found that boron will affect not only the density of TD but also the shape of them. Finally, the TDD values from EPD and PVTEM are summarized in Table 3.6.  Sample  TDD value (cm-2)  Sample  TDD value  (cm-2)  Sample  TDD value (cm-2) U-No TC 3.5 ± 1.5 × 106  A-No TC 5 ± 3 × 105  LB-5TC 1.5 ± 0.5 × 108 U-5TC < 5 × 104  A-5TC 1.2 ± 0.5 × 105  LB-HT 1.6 ± 0.5 × 108 P-No TC 3 ± 1 × 105  A-HT 1.2 ± 0.5 × 105  HB-No TC 8 ± 2 × 108 P-5TC 1.75 ± 1 × 105  LB-No TC 4.5 ± 1.5 × 108  HB-5TC 1.7 ± 0.5 × 108 Table 3.6 TDD values of the 12 samples measured by EPD, SEM (bold) and PVTEM (Italics).   Cross section TEM (XTEM) has been performed to detect the TDD levels in the Ge seeding layer. Due to its requirement for extensive and skillful sample preparations, we (a) (b) 64  have only detected the sample U-5TC and LB-5TC, which represent the samples with low/high TDD in the Ge bulk layer respectively. Figure 3.10 shows the XTEM images of sample U-5TC and LB-5TC respectively. We can see that the TDs mainly lays in the Ge side closing to the Si-Ge interface, namely, the Ge seeding layer (~100 nm thickness). Despite the fact that U-5TC has a much lower TDD than LB-5TC in the Ge bulk layer, both samples have an extremely high density of TDs in the Ge seeding layers, which are estimated to be over 1 × 109 cm-2. These massive misfit dislocations were generated on the boundaries during annealing to relieve the tensile strain between layers, as shown in HRXRD results in Section 3.6. Boron dopant doesn't have much impact on the TDD in the Ge seeding layer according to the comparison between the two samples. Considering the fact that U-5TC has the lowest TDD in Ge bulk layer and LB-5TC has a much higher TDD than U-5TC, P-5TC and A-5TC, it is reasonable to assume that the TDDs of P-5TC and A-5TC in Ge seeding layer should also be in similar range. Thus, we conclude that all dopants had little impact on the TDD level in the Ge seeding layer, where the Si-Ge interdiffusion mainly happened.  65   Figure 3.10 Cross section TEM images in bright mode of Sample (a) U-5TC and; (b) LB-5TC. The TDD levels in Ge seeding layer of both samples are estimated to over 1× 109 cm-2.  3.5 Atomic Force Microscope (AFM) Measurements AFM has been widely used as an effective surface morphology tool in the study of semiconductors. This method has been applied to characterize the surface roughness of the samples. The scanning size was chosen to be 1 μm × 1 μm and at least three different sites have been selected to scan for every sample. As an example, Figure 3.11 illustrates two results from sample U-No TC.  According to our experiments, we find there exist lots of bumps which are around of 0.2 μm diameter and 10 nm height. Those randomly distributed bumps will severely affect the roughness calculation results. Take the results sample U-No TC as an example, the average roughness in Figure 3.11 (a), which is with no bump, is 0.379 nm. As a comparison, that in Figure 3.11 (b), which contains some bumps, is 2.01 nm. Considering the effect of those bumps, we selected areas without any bump Ge Si Si Ge (a) (b) 66  to perform another roughness calculation. The final results are the average of the two calculations, which are listed in Table 3.5.      Figure 3.11 Examples of AFM measurement results from Sample U-No TC, P-No TC and P-5TC.  67  Those bumps should be due to the Ge sublimation during the growth/annealing. That Ge will even sublimate at low temperature (~600 oC) has been reported by several group [151-154]. Similar curve feature containing bumps has also been observed [152]. However, the mechanism behind that and the sublimation rate are still under debate. The sublimation of Ge layer for the annealing samples in our experiment is roughly estimated to be 20-70 nm. Furthermore, according to the AFM results for different samples, neither the annealing procedure nor different dopants configuration will have any impact on that. Table 3.6 lists the surface roughness calculation results. There are two different calculation methods. One is Average Roughness (Ra), one of the most commonly used roughness statistics. It is the arithmetic average of the absolute values of the surface height deviations measured from the mean plane, expressed as: 𝑅𝑎 =1𝑁∑ |𝑍𝑗 |𝑁𝑗=1                                                                                                  (3.1).  Another roughness values are based upon least-squares calculations (e.g., RMS roughness, or Rq), and their algorithms are more concerned with the best fit of all height points than with the spatial frequency of features. It is an average of height deviations taken from the mean image data plane, expressed as:   𝑅𝑞 = √∑ |𝑍𝑗|𝑁𝑗=1𝑁                                                                                                    (3.2).  Besides, a plane fitting procedure shall be performed before the roughness calculations, as discussed in Section 2.2.4. The higher order the plane fitting is, the smoother the final plane is. We chose the second order plane fitting, and its corresponding polynomial is listed as follows:  68  z = a + bx + cy + dxy + ex2 + fy2                                                                        (3.3), where a-f are all fitting coefficients.  Average roughness (nm) RMS roughness (nm) U-No TC 0.65 ± 0.15 0.91 ± 0.26 U-5TC 0.59 ± 0.20 0.80 ± 0.29 P-No TC 0.42 ± 0.08 0.57 ± 0.14 P-5TC 1.05 ± 0.35 1.49 ± 0.48 A-No TC 0.34 ± 0.05 0.32 ± 0.03 A-5TC 0.28 ± 0.01 0.35 ± 0.01 A-HT 0.82 ± 0.29 1.43 ± 0.69 LB-No TC 0.27 ± 0.04 0.34 ± 0.05 LB-5TC 0.28 ± 0.03 0.36 ± 0.03 LB-HT 0.31 ± 0.03 0.39 ± 0.04 HB-No TC 0.52 ± 0.17 0.78 ± 0.35 HB-5TC 0.66 ± 0.25 0.97 ± 0.45 Table 3.7 The average and RMS surface roughness of samples after the second order of plane fitting.  As shown in Table 3.7, although the average roughness for different samples may vary from 0.2 nm to 1.0 nm, we still think they are in the same range considering the fact that even the results of the same sample can also vary within that range (for example, the sample in Figure 3.11). In conclusion, different dopant configurations or annealing procedures have no significant effect on the surface roughness. All samples have similar surface quality. That smooth surface is eligible for the transition layer between III-V layer and Si layer.         69  3.6 High-Resolution X-ray Diffraction (HRXRD) Results HRXRD has been used for decades to investigate semiconductor structures. It has some unique features, including its nondestructive character, a good match of X-ray wavelength to the atomic scale of modern semiconductor devices and the rapid collection of statistically significant data [155]. X-ray diffraction is one of the two techniques to obtain Ge fraction and strain in Si-Ge layers simultaneously, which is important for the embedded Si-Ge layers in complicated structures like the samples used in this thesis. [116] HRXRD has been applied to measure the Ge strain level of the samples. The measurements were performed in Advanced Materials and Process Engineering Laboratory (AMPEL) at UBC using a PANalytical X’Pert PRO MRD with a triple axis configuration.  3.6.1 Kinematical theory underlying XRD The interaction of X-rays with matter happens through the scattering by the electrons of the atomic constituents. Scattering is most easily understood by thinking of a plane wave. When such a plane wave strikes a three-dimensional atomic lattice, each scattering point (electron or nuclear particle) acts as a source of spherical waves, whose wavefronts lie on spheres centered on the scattering points [156]. The simplest and most useful description of crystal diffraction is still that obtained by Bragg. Strong diffraction occurs when all the wavelets add up in phase. By considering an entire crystal plane as the scattering entity, rather than each individual electron, it is easily seen that the strong diffraction results happen when Ea. (3.4) is satisfied 70  2𝑑𝑠𝑖𝑛𝜃 = 𝑛𝜆                                                                            (3.4) The intensities of the diffracted beams depend upon the strength of the scattering that the material inflicts on the radiation. The basis scattering unit of a crystal is its unit cell, and we may calculate the scattering at any angle multiplying the scattering strength of one electron or nucleus, then that of an atom, next that of a unit cell and finally that of the total number of unit cells. The amplitude of the wave scattered by a crystal can be expressed as [157]: 31 21 2 311 120 1 1 2 2 3 320 0 001( ) exp( ( ))NN Ncrystn n neA A F q iq n a n a n amc R        ,                     (3.5) Where A0 is the amplitude of the incident wave, e, m, c are the charge of an electron, mass of an electron and the light speed in vacuum, 𝑒2𝑚𝑐2 is called Thomson scattering length; R0 is the observation distance. N1, N2, N3 are the numbers of unit cell along the vector directions 1a , 2a , 3a . f lq k k   is the momentum transfer and in the case of elastic scattering 2 /f lk k    , and 31( ) ( )exp( )cNj jjF q f q iq r d r  is the structure factor of the unit cell and Nc is the number of atoms in the unit cell, defined by the vectors 1a , 2a , 3a . jr  is the position vector of an atom in the unit cell. All the atoms of the unit cell have their own atomic scattering factors ( )jf q .  The structure factor F(q) is a fundamental quantity, which appears in all the expressions for diffracted intensity, penetration depth and rocking curve width. This factor is calculated under some ideal assumptions: the scattered intensity is minimal (indicating the loss of intensity due to re-scattering is negligible); R0 is large compared 71  with the dimensions of any coherently illuminated scattering volume; and scatted waves from different atoms are nearly parallel [156]. The atomic scattering factors ( )jf q  are usually calculated in terms of scattering of an individual free electron. This is calculated as if the electron were a classical oscillator since the assumption is that the electron is a free charged particle. It is set into a forced oscillation by the radiation field of an incident X-ray and then re-radiates in all directions at the same frequency as the incident wave frequency, which is called elastic or Thompson scattering.  3.6.2  HRXRD results and discussion HRXRD is used to confirm the degree of strain relaxation and lattice constant of Ge with (0 0 4) Ω–2θ scans. The position of the Ge peak coincides with the ideal Ge (0 0 4) position. The perpendicular lattice constants measured from the Ge peak positions samples are around 5.66708 Å. The associated degrees of relaxation R calculated by combining Eqs. (3) and (4) from Ref. [158] are 104%. Details of the calculations and simulations are attached in the Appendix B. This means that the Ge layers are in a slightly tensile strained configuration (~ 0.16%), which is in agreement with the results reported by Hartmann et al. [158][159] and MIT researchers [160]. The tensile strain is thermally induced in the Ge epilayer during cooling from high-temperature processing steps to room temperature, as Ge has a linear coefficient of thermal expansion (CTE) of 5.8 ppm/oC comparing to Si of 2.6 ppm/oC [161]. The (0 0 4) Ω–2θ scans results of the samples are shown in Figure 3.12. The analyses for the unannealed samples demonstrate that the relaxation R are all within the 72  range 103.9% - 105%. The slight difference between different samples should be due to doping impact on the lattice constant, as P and B atoms size are smaller Ge while As is larger than Ge. Besides, the Ge film tilt is also slightly different for different  samples, which can also lead to the Ge peak shift. For the samples after annealing, the XRD results are more complicated. Compared with the HRXRD results of the samples before annealing, the Ge peaks of the samples after annealing are much more asymmetric. They are wider towards the high incidence angle side. This is due to SiGe interdiffusion during the annealing, forming a SiGe alloy region with a graded Ge concentration. This is consistent with the SIMS results in Section 4.1.1. Sample P-5TC and A-5TC have larger interdiffusion than other samples, making the Ge peaks more broadened. The interdiffusion also shifts the “Ge” peaks towards the Si side, as the “Ge” bulk layers are no longer uniform Ge layers with 100% Ge. For example, according to the SIMS data, at the surface of P-5TC, 𝑥𝐺𝑒  = 0.98, and 𝑥𝐺𝑒  decreases more with the depth.  In this case, it will be inaccurate to calculate the exact relaxation R of the Ge layers solely through the (0 0 4) scans. The Ge concentration information should also be included. From the XRD “Ge” peak, the relaxation of this layer can be calculated as 105.9% considering it as 𝐺𝑒0.98𝑆𝑖0.02 . Considering the fact that 𝑥𝐺𝑒 decreases with the depth, the relaxation of all other regions is even less than 105.9%. Thus, we can conclude that for P-5TC, the Ge layer became a SiGe alloy region with the surface Ge concentration being 𝑥𝐺𝑒  = 0.98. This SiGe alloy region is still almost entirely relaxed. The same argument can be applied to A-5TC as its 𝑥𝐺𝑒  at the top is around 98.5%. The bottom line is that in the annealed samples, the top layer is a Ge rich layer with similar slight tensile strain, 73  which is not going to influence the interdiffusion significantly.   Figure 3.12 HRXRD results of the samples (a) without annealing; and (b) after annealing. The results show that the Ge layers are almost fully strained relaxed.   To further confirm our results, (2 2 4) reciprocal space maps have been performed for sample U-No TC and P-No TC as illustrated in Figure 3.13. The results are consistent with the conclusion from (0 0 4) Ω–2θ scan.  74   Figure 3.13 HRXRD result of (2 2 4) reciprocal space mapping of Sample (a) U-No TC; and (b) P-No TC.  In conclusion, HRXRD results demonstrate that all samples are nearly entirely strain-relaxed. Dopants have no significant influence on the strain degree of the Ge films.   3.7 Analysis and Chapter Summary In summary, this chapter focuses on the impact of doping with As/P/B on Ge-on-Si film quality. 12 Samples with 5 different doping configurations (U, A, P, LB and HB) and 2 different annealing conditions have been designed and annealed. EPD, SEM and TEM measurements have been performed to detect the TDD values in the Ge bulk layers and the Ge seeding layers of different samples. The results are summarized in Table 3.8. It is found that boron introduces lots of extra TDs in the Ge bulk layer, making it unsuitable for electronic or photonic devices. As and P can reduce the TDD level in unannealed samples greatly. This provides a new method to fabricate high-quality Ge-on-Si films without defect annealing procedure, which can avoid undesired Si-Ge 75  interdiffusion. This is one of the major findings of this work. AFM and HRXRD have been performed to measure the surface roughness and the strain levels respectively. All samples have a smooth surface (roughness < 2nm) and the Ge layers in annealed samples are nearly fully relaxed. None of the dopants have any significant impact on the surface roughness and the strain values.    Methods  Purpose  U P As LB  AFM Surface Roughness 0.5~1.5 nm HRXRD Ge film strain ~104% relaxed, small tensile strained due to thermal expansion mismatch EPD & SEM before annealing TDD in bulk Ge at ~ 300 nm depth (cm-2) 3.5× 106 3  × 105 5 × 105 > 2 × 108 EPD & SEM after annealing TDD in bulk Ge at ~ 300 nm depth (cm-2) < 5 × 104 1.75  × 105 1.2  × 105 1.2 × 108 TEM TDD all depth < 1 × 107 (too low for PVTEM) 4 × 108 Table 3.8 Summary of the Ge films quality measured by AFM, HRXRD, EPD, SEM and TEM.  76  Chapter 4: Impact of Doping on Si-Ge Interdiffusion In this chapter, we first present the Ge and dopant profiles for the samples before and after annealing as measured by SIMS. As a preliminary analysis, the Boltzmann-Matano method was used to extract the time-average Si-Ge interdiffusivity as a function of the Ge molar fraction (𝑥𝐺𝑒). By comparing sample U-5TC and other well-established models, the best-fitting annealing temperatures of sample A-5TC and sample P-5TC were determined. Finally, by modeling and fitting the Ge profiles after annealing, a quantitative model of Si-Ge interdiffusion under n-type doped was established.   4.1 Experiments 4.1.1 SIMS results and discussions The following figures show the Ge profiles and dopants profiles in the samples with annealing and without annealing, respectively. It is worth mentioning that the profiles from different samples have been shifted laterally. The reasons for doing that are as followings: 1) during the epitaxial growth, a ± 3% cross-wafer thickness non-uniformity is typical, which shifts the Ge profile laterally and 2) the SIMS depth measurement uncertainty is about ± 30 nm, which is close to the interdiffusion length (10 - 60 nm) of the samples in this study. Therefore, one cannot use the absolute depth of the SIMS profiles due to the thickness non-uniformity and depth errors discussed above, even if the two samples are taken from the same wafer side by side. To compare the amount of interdiffusion, we use the slope of the Ge profiles as the evaluation criteria. Steeper Ge profiles mean less interdiffusion and vice versa. 77  Figure 4-1 shows the Ge and dopants (As/P/B) profiles in the samples without annealing. According to the figure, all samples have very similar sharp Ge profiles at the Ge/Si interfaces before annealing. The Ge profiles almost overlap with each other. All the dopants have the highest concentration at the interface of Ge/Si, which is due to the segregation induced by a high density of defects at the interfaces. The maximum concentration of As and HB are around 2.8 × 1020 cm-3, and that for P and LB are 1.5 × 1020 cm-3 and 8 × 1019 cm-3 respectively. Besides, As and P have a higher concentration in the bulk Si layers than in the bulk Ge layers away from the interfaces, while B has a different preference, which is due to dopant segregation between bulk Ge and Si. It is also worth mentioning that all the dopants concentrations are below their corresponding solubility in Ge and Si at T = 850 °C [111][162], which is the main thermal budget in the annealing process. Therefore, there is no second phase formation. Different dopants have led to different interdiffusion profiles after annealing in spite of their similar Ge profiles at the beginning. Figure 4.2 compares the Ge profiles in the samples with and without annealing. The dashed lines stand for Ge profiles of the samples without annealing, and the solid lines stand for those with annealing.  78   Figure 4.1 Ge and dopants (As/P/B) profiles of samples without annealing measured by SIMS.  The dashed lines are the Ge profiles. The solid lines are the dopant profiles. The Ge profiles are shifted laterally for easy comparison. The dopant profiles are also shifted laterally by the same length as their corresponding Ge profiles. HB/LB/A/P/U stand for high concentration B doped/ low concentration B doped/ As doped/ P doped/ undoped samples respectively.    Before annealing, all samples have similar sharp Ge profiles at Ge/Si interfaces. From Figure 4.2, we can see that sample U and sample LB have the least interdiffusion while sample P has the largest. Sample A has the second largest interdiffusion. While for sample HB, it has no significant difference from sample LB in 𝑥𝐺𝑒  < 0.7 part, but it distinguishes itself from LB and U in 𝑥𝐺𝑒  > 0.7 part. The interdiffusion profiles show a strong 𝑥𝐺𝑒  dependence, where much more diffusion happens in high Ge regions than in low Ge regions.  The interdiffusion regions for P and U are 620 nm and 320 nm thick respectively if we define the interdiffusion region is from 0.02 < 𝑥𝐺𝑒< 0.98. They are 570 nm and 170 nm 79  thick if we define the interdiffusion region as 0.05 < 𝑥𝐺𝑒  < 0.95. The time-averaged interdiffusivity was extracted by the Boltzmann-Matano analysis and discussed in Section 4.2.  Figure 4.2 Ge profiles of samples with and without annealing measured by SIMS.  The dashed lines are the Ge profiles of samples without annealing. The solid lines are the Ge profiles of samples with annealing. The Ge profiles are shifted laterally for easy comparison. HB/LB/A/P/U stands for high concentration B doped/ low concentration B doped/ As doped/ P doped/ undoped samples respectively.   Figure 4.3 dopants (As/P/B) profiles of samples with and without annealing measured by SIMS.  80   Different dopants have also shown different changes after annealing. Figure 4.3 compares the dopants profiles in samples with and without annealing. According to the figure, As and P still have peak concentration in the Si/Ge interface after annealing while B doesn't have that, which is attributed to the dislocation density differences and will be analyzed in Session 4.2. Comparing the dopant profiles in samples with and without annealing, we can easily find that dopant concentrations don ’t change much on the Si side, but they have increased significantly on the Ge side, which results from the faster diffusion coefficients in Ge than those in Si. The dopants intrinsic diffusion coefficients at T = 850 °C are listed in Table 4.2 [163-165]. Besides, the concentration preferences for dopants in Ge/Si still hold as illustrated in Figure 4.3. As and P prefer to concentrate in the Si layer, while it’s opposite for B. Except for Sample LB, their concentrations are higher than the corresponding intrinsic carrier density (𝑛𝑖) in Si (2.4 × 1018 cm-2) [166] and Ge (1.1 × 1019 cm-2) [166] at T = 850 °C. Although P and As have a clear concentration gradient in the Ge side compared with the beginning condition, we should realize that it takes a much shorter time than our annealing time (50 minutes) for As and P to reach a state close to their equilibrium state.   DAs (cm2/s) DB ( cm2/s) DP ( cm2/s ) Si 1 × 10-17 3.9 × 10-16 2 × 10-16 Ge 2.2 × 10-11 1.5 × 10-16 1.1 × 10-12 Table 4.1 Intrinsic diffusion coefficients of different dopants (As, B and P) in Si/Ge at T = 850 °C [163-165].    81  4.1.2 The impact of different annealing procedures   Figure 4.4 Ge and dopants profiles in samples without annealing (w/o annealing), with 5TC and HT  (a) As doped samples; and (b) low concentration B doped samples.   As  illustrated in Figure 4.4, no significant difference has been observed between samples annealed by 5× thermal cycle (5TC) and those with merged high-temperature annealing (HT), which indicates that it is the high temperature (850 °C) annealing thermal budget that determines the interdiffusivity results. Furthermore, we compared extracted time-average interdiffusivities using the Boltzmann-Matano method, as shown in Figure 4.5. The result is consistent with their corresponding Ge SIMS profiles. These two different annealing procedures (5TC and HT) have a negligible distinction as long as their high temperature (850 oC) annealing time (50 minutes) are the same. This is expected as interdiffusivity has a strong temperature dependence in Arrhenius form. The tramps and the low-temperature thermal budget are negligible compared to the high-temperature thermal budget.   82   Figure 4.5 Comparison between the extracted time-average interdiffusivities of different samples annealed under different thermal conditions. The lines without markers A-5TC and LB-5TC. The lines with markers A-HT and LB-HT.  We also compared those samples in the materials characterization results in Chapter 3 using EPD, AFM, TEM, and HRXRD. The results are consistent with the SIMS results that those two annealing conditions have little impact on diffusion, interdiffusion or Ge film quality.   4.2 Quantitative Modeling of Si-Ge Interdiffusion with Doping In this part, we will quantitatively analyze the Si-Ge interdiffusion behavior under doped conditions. Firstly, we extracted the effective interdiffusivity using the Boltzmann-Matano method and determine the annealing temperatures by comparing results with literature studies. Then, attributing the enhancement of interdiffusion to Fermi level effect, we model the interdiffusion and simulated Ge profiles after annealing to validate the model. 83    4.2.1 Extraction of Si-Ge interdiffusivity In the following section, Boltzmann–Matano analysis was used to extract the time-averaged effective interdiffusivity (𝐷𝑆𝑖−𝐺𝑒) as a function of the Ge faction (𝑥𝐺𝑒) from the concentration profiles. Before we discuss the extraction results, it’s worth investigating the validity of the Boltzmann-Matano method in this experiment. For this analysis method to be valid, the interdiffusivity of Ge-Si needs to be a function of the concentration or molar fraction of Ge (or Si) only. In Xia et al.’s analysis [93], a fully strained and undoped SiGe epitaxial layers were grown on Si substrates with a low dislocation density ( < 105 cm-2). As the biaxial strain dependence can be expressed as the 𝑥𝐺𝑒  dependence and the defect concentration is negligible, the Si-Ge interdiffusivity is a function of  𝑥𝐺𝑒  only, which satisfies the condition of the Boltzmann-Matano analysis. However, for samples in this work, the condition that Si-Ge interdiffusivity is a function of 𝑥𝐺𝑒  only could not be fully satisfied. As we know, interdiffusivity depends on stress status, defect concentration, and dopant concentration. For the material systems in our study, due to the non-uniform defect concentrations and dopant concentration vs. depth and the time evolution of these two concentrations, the Boltzmann-Matano analysis condition is not met.  84   Figure 4.6 Ge and dopant (As/P) profiles of the samples after annealing and without annealing. The dashed lines the profiles of the samples without annealing. The solid lines are the profiles of the samples after annealing. The Ge profiles are shifted laterally for easy comparison. The dopant profiles are also shifted laterally by the same length as their corresponding Ge profiles.   However, we can still use the Boltzmann-Matano method to extract the effective interdiffusivity (time and position averaged interdiffusivity) in our experiments due to the following reasons: 1) The change from the as-grown P/As profiles to the annealed profiles are not significant in most regions of interest as shown in Figure 4.6; and 2) P and As diffuse much faster than Si-Ge interdiffusion. P and As diffusivity in Ge are over 20 times larger than 𝐷𝑆𝑖𝐺𝑒  [94][167]. Therefore, we consider that most of the P and As motion happen in the early stage of the annealing process, and reach a distribution close to the final distribution, while for Si-Ge interdiffusion, as it is much slower and further away from the equilibrium, we expect that the interdiffusion motion happens throughout the annealing. From that logic, in the annealing process, we expect that P and As reach a 85  relative stable state much faster, and continue with small changes afterwards. Most of the interdiffusion motion happens after P and As reach a relative stable state, and we can approximate that the interdiffusion happens with a fixed P and As distribution same as the final profiles.  To confirm this assumption, we annealed the P doped sample at T = 870 oC for 50 minutes and 200 minutes respectively. This experiment was done by the author using a heating stage at UBC. The SIMS results of the annealing results are shown in Figure 4.7. The shift of P profile from 50 minutes annealing to 200 minutes annealing is less than 60 nm, which is due to the Kirkendall effect and also thickness nonuniformity and SIMS errors.   Figure 4.7 Ge and P profiles of P doped samples with 50 minutes annealing and 200 minutes annealing respectively. The annealing T = 870 oC.  In conclusion, for Sample P and Sample A, we can approximate that the dopants have reached their final state sooner after the annealing began and extract the 86  interdiffusivity as a function of 𝑥𝐺𝑒 . This doesn’t mean there are no time or position dependence. During the annealing, although the P and As profiles stabilize fast, the Ge profiles change at the interface, where the defect concentration depends heavily on the position. We also expect to see the Kirkendall effect where more Si lattice sites come to the Ge side due to much faster Si intrinsic diffusivity, which may cause the movement of the interface and dopant profiles. Although these effects are inevitable, with many other parameters unavailable such as the dopant and defect dependence of the interdiffusivity, so far the Boltzmann-Matano method is still the best method in giving a rough estimate of the interdiffusivity. The definition of the effective interdiffusivity is such that when this interdiffusivity is used, the experimental Ge profiles after annealing can be reproduced in numerical simulations using Fick’s laws. For Sample LB and HB, although the conditions discussed was no longer satisfied due to its diffusion in Ge (1.5 × 10-16 cm2/s) [163] is comparable to Si-Ge interdiffusivity, we could still use this method to extract the interdiffusivity as a reference since B shouldn’t have much impact on interdiffusivity according to the SIMS profiles.  4.2.2 Interdiffusivity extraction results and discussions  The extracted interdiffusivity was illustrated in Figure 4.8. The results are consistent with their corresponding Ge profiles. 87   Figure 4.8 The time-averaged interdiffusivity as a function of Ge fraction using the Boltzmann-Matano method extracted from Sample U/P/A/HB/LB. The major thermal budget is the HT anneals at 850 °C. The annealing time is 50 minutes.  Sample P has the highest interdiffusivity in the full 𝑥𝐺𝑒  range and the Sample A has the second largest interdiffusivity. The interdiffusivity of Sample P (𝐷𝑃) is 1.5~3 times higher than that of sample A (?̃?𝐴), and 𝐷𝐴 is 1.5~2 times higher than that of Sample U (𝐷𝑈). Both As and P will enhance the interdiffusivity, which agrees with the previous study from our group [107][108].  On the other hand, the Sample LB, Sample HB, and Sample U do not exhibit much difference, especially in the 𝑥𝐺𝑒  < 0.6 part. In the 𝑥𝐺𝑒  > 0.6 range, their inter-diffusivities do have some differences and it shows that  𝐷𝐻𝐵 > ?̃?𝑈 > 𝐷𝐿𝐵. However, we should also keep in mind that we shouldn’t over-interpret the difference, considering the SIMS broadening effect and the data resolution limiting. The difference is even less than the 88  error bar of our data extraction, which is 50%. Besides, the Boltzmann-Matano approximation is not valid with B doping, as its diffusion in Ge (1.5e-16 cm2/s) [163] is comparable to the Si-Ge interdiffusivity. Furthermore, it is found that the extracted inter-diffusivities can be very sensitive to calculation details like the smoothing calculation, which will be discussed in detail in the following section. The SIMS profiles of Sample U, HB and LB have also indicated that they had little difference in interdiffusion.  To conclude, n-type doping (As and P) can enhance the Si-Ge interdiffusivity significantly while boron’s effect on that is small if any. It is also worth noting that the n-type dopant introduced enhancement depends on the doping level. Since the P doping level is higher than As, we cannot conclude whether this enhancement observed is related to the doping level or to the doping level and n-type dopant species.  4.2.3 Effects from SIMS data smoothing and extraction To check the reliability and accuracy of the Boltzmann-Matano method, we will discuss some data processing and extraction details below. The analytical expression for interdiffusivity extraction by the Boltzmann-Matano analysis is given by Ea. (4.1): 𝐷(𝐶′) = −12𝑡(𝑑𝑧𝑑𝐶)𝐶=𝐶′∫ (𝑧 − 𝑧𝑀)𝑑𝐶𝐶′𝐶𝐿,                                                                      (4.1) ∫ (𝑧 − 𝑧𝑀)𝐶𝑅𝐶𝐿𝑑𝐶 = 0,                                                                                                       (4.2) where 𝐷(𝐶′) is the interdiffusivity at Ge concentration 𝐶′, t is the annealing time, z is the depth, 𝑧𝑀 is the position of the Matano plane. The definition for Matano plane 𝑧𝑀 should satisfy Equation (4.2), where 𝐶𝑅 and 𝐶𝐿 are two constant concentrations (R and L denote for right and left) on the two sides of the interdiffusion couple. The original SIMS data 89  inevitably contained some random noise. Such noise is amplified in the following calculations and may lead to some unreasonable results, which mainly originate from the 𝑑𝑧𝑑𝐶 calculation. Especially, in the 𝑥𝐺𝑒  →  0 and 𝑥𝐺𝑒 →  1 region, the 𝑑𝐶  between two adjacent data points can be positive instead of negative due to the data noise. This results in a negative interdiffusivity, which is obviously incorrect and were discarded in our diffusivity extraction. Proper smoothing is needed before we can use the Boltzmann-Matano method. One of the most common algorithms is the "moving average" as explained in Section 2.3.1.  On the other hand, too much smoothing will make the profile less “abrupt” than the real data in the region where Si-Ge interdiffusion mainly happens, and will artificially increase the interdiffusivity, too. It is worth noting that similar is true for the SIMS broadening effect, which makes the slopes less abrupt, and increases the interdiffusivity artificially. Some compromise is needed to balance the two requirements.  In Figure 4.9 (a), only part of SIMS data (𝑥𝐺𝑒  < 0.1 and 𝑥𝐺𝑒  > 0.9) was smoothed while data in all 𝑥𝐺𝑒   range has been smoothed in Figure 4.9 (b). Although no significant difference has been observed, the latter smoothing method is still not favored due to the error of smoothing in the slope region. On the other hand, improper smoothing can also lead to artificial results. In Figure 4.9 (c), the interdiffusivity data was smoothed after being extracted from the SIMS data. Obviously, the result in Figure 4.9 (c) is inaccurate since the interdiffusivity does not decrease with an increasing Ge faction, let alone interdiffusivity in Sample U has the highest value in 𝑥𝐺𝑒  < 0.4 part, which apparently contradicts with the SIMS Ge profiles. Therefore, we used the “part of SIMS data smoothing method” as shown in Figure 4.9 (a). 90    Figure 4.9 The extracted time-average interdiffusivity using the Boltzmann-Matano method with different smoothing methods; (a) the SIMS data was only  smoothed in the range 𝒙𝑮𝒆< 0.1 and 𝒙𝑮𝒆 > 0.9; (b) the SIMS data in all 𝒙𝑮𝒆 range has been smoothed; (c) interdiffusivity was smoothed after extracting.  4.2.4 Annealing temperature extraction It is worth comparing the extracted interdiffusivity of Sample U (undoped) in this work with two relevant studies. Gavelle et al. [101] studied the interdiffusivity of the highly defected Ge layer on Si substrate with a dislocation density of about 1010 cm−2, which is similar to the Ge seeding layer of our Sample U. Dong et al. [94] established a thermodynamic model for Si–Ge interdiffusion with a low defect density (about 105 cm−2) without doping. Although the annealing in [94] and [101] was performed in an inert ambient, and our samples were annealed in a hydrogen environment, no significant 91  difference in Si–Ge interdiffusion was reported for the anneals with two different gases [107][108]. The major thermal budget in this work was the HT anneals at 850 °C, which is different from the temperatures used in their work. To compare the results, we calculated the interdiffusivities at 850 °C using models from Gavelle et al. and Dong et al.’s papers, as shown in Figure 4.10 (a).  Figure 4.10 Time-averaged interdiffusivity from Sample U (undoped) in comparison with the interdiffusivity calculations from literature models in [101] [94]; (a) literature model at 850 oC; (b) literature model at 890 oC.  From the SIMS data in Figure 4.2, we can see that in Sample U, the interdiffusion happens mainly in the 100 nm thick Ge seeding layer. According to the TEM analysis in Chapter 3, dislocation density in Ge seeding layers are larger than 1010 cm−2. There is a high density of point defects in the seeding layer. Considering the dislocation density differences, the interdiffusivity of Sample U should be equivalent to that in Gavelle et al.’s result and be higher than Dong et al.’s result. Indeed, from Figure 4.10 (a) we can see that, at the Si end, the time-averaged effective interdiffusivity extracted from Sample U is 3 × 10−16 cm2 s−1, which is three orders of magnitude higher than the low-defect case and 2 92  times larger than the Gavelle et al.’s samples. At the Ge end, the difference between the three interdiffusivities curves is within a factor of three.  The 3X difference is within the normal accuracy range of common diffusivity studies. The error bar of the previous interdiffusion studies from our group and Dr. Xia’s studies at MIT is common within a factor of 2X due to more accurate SIMS and careful temperature calibration [93][94][116][108]. For (inter)diffusion studies, the calibration of temperatures is essential to the accuracy of (inter)diffusion modeling. Commonly used annealing tools include furnace, heating stage, CVD, and rapid thermal annealing (RTA) equipment. The former three methods use resistive heating while the last one uses radiative heating. Therefore, the former three are much easier to calibrate and more repeatable. In this study, the annealing was performed inside the MOCVD growth tool. That inconsistency with Gavelle et al.’s result probably originates from the differences in temperature calibrations. As discussed in Section 3.1, the annealing was performed in the MOCVD tool with 2 different temperature monitors in this work. One is a thermal couple below the susceptor to set the heating power, and another is a multi-channel pyrometer to read the wafer surface temperature that allows real-time surface temperature measurement. Usually, the wafer surface temperature is about 100 – 150 oC lower than the thermal couple temperature. Especially, the surface temperature of the As doped sample was always about 20 oC slightly lower than the other specimens, even though they were targeted at the same thermal couple temperature setting. In this experiment, all the temperatures quoted so far have been the surface temperature 93  readings. The reading As the pyrometer was calibrated 2 years ago, it was very likely that some errors existed in the pyrometer temperature readings.  In judging which temperature is more correct, we define the correct one should be consistent with literature data in this area that used different annealing tools and with various temperature measurements and calibration methods. It should be pointed out that Dong et al.’s model is a benchmarking model in this area. The model was based on experimental data with no doping effect and very low defect density. Thus, his model fits the lattice interdiffusion term quite well. It agrees with the majority of the experimental data in the area [94]. Gavelle et al.’s data were measured on samples with high dislocation density similar to our case. In the high Ge regime, as the lattice interdiffusion term dominates, it agrees with Dong et al.’s model well, which shows that the temperature calibration of Gavelle et al.’s data is consistent with Dong et al.’s model. Therefore, if we use Gavelle et al.’s model and a temperature to fit our experimental data, this best-fitting temperature is the extracted temperature, which is better as it agrees with the majority temperature calibrations in the field.   Following the above logic, we find that Gavelle et al.’s model at 890 oC fits our effective interdiffusivity of Sample U very well as illustrated in Figure 4.10 (b). Therefore, 890 oC is the extracted temperature in previous interdiffusion models and data discussed (pyrometer temperature reading as 850 oC).  Furthermore, interdiffusion can be mediated both by threading dislocations and by point defects in the lattice as seen in Ea. (4.3). In [101], the dislocations-mediated term D̃dislocation  was modeled to have a weaker Ge dependence than the point-defects-94  mediated term D̃lattice as illustrated in Ea. (4.4) and Ea. (4.5), where k is the Boltzmann constant and T is the temperature in Kelvin.   𝐷𝑡𝑜𝑡𝑎𝑙 = 𝐷𝑑𝑖𝑠𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 + 𝐷𝑙𝑎𝑡𝑡𝑖𝑐𝑒                                                                         (4.3)  ?̃?𝑑𝑖𝑠𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 = 10−2 𝑒𝑥𝑝(2.5𝑥𝐺𝑒) ∗ 𝑒𝑥𝑝 (−3.1𝑒𝑉𝑘𝑇)                                                  (4.4)  ?̃?𝑙𝑎𝑡𝑡𝑖𝑐𝑒 = 1.12 × 10−4𝑒𝑥𝑝(12.75𝑥𝐺𝑒) ∗ 𝑒𝑥𝑝 (−3.33𝑒𝑉𝑘𝑇)                                     (4.5) In Gavelle et al. work, both terms have been modeled with Arrhenius terms, and it is mathematically reasonable. In Dong et al.’s model, the dislocation density is rather low (about 105 cm−2). Thus, the interdiffusivity is in Dong et al.’s model is almost entirely the point-defects-mediated interdiffusion term. It is worth comparing 𝐷𝑙𝑎𝑡𝑡𝑖𝑐𝑒 in Gavelle et al.’s model with Dong et al.’s model at temperature T = 850 oC/890 oC. As illustrated in Figure 4.10 (a) and (b), 𝐷𝑙𝑎𝑡𝑡𝑖𝑐𝑒  in Gavelle et al.’s model is in consistent with Dong et al.’s model both at T = 850 oC/890 oC, which further validated their hypothesis. Based on the above facts and considerations, we believe that a more accurate annealing temperature should be 890oC instead of 850 oC for Sample U/P/LB/HB. For the sample A, since the surface temperature was about 20 oC lower than others, we believe the extracted annealing temperature should be 870 oC.   4.2.5 Mechanisms of interdiffusion enhancement From the experimental results above, we can see that as long as high P/As doping exists, the Si–Ge interdiffusivity will increase drastically. To interpret this phenomenon, we need to discuss a few possible mechanisms: 95  1) Defect density. The possibility that this enhancement is due to a defect density difference is quite unlikely as the TDD values of Sample U and LB are close at Ge/Si interfaces. It is reasonable to assume that Sample P/A exhibit a similar result as discussed in Section 3.4.  2) Strain. From our XRD results, Ge layers of all samples are under around 0.16% tensile strain from the coefficient of thermal expansion (CTE) mismatch. From the epitaxial growth point of view, the Ge layers are much thicker than the critical thickness, and therefore, the Ge layers are fully relaxed except for the CTE mismatch. There were no other sources to introduce strain difference. Therefore, all samples with different doping have similar strain levels, and the enhanced interdiffusion cannot be a result of strain difference. 3) Fermi-level effect. It is known that when a dopant concentration is close to or higher than the intrinsic carrier concentration 𝑛𝑖, the Fermi-level effect can change the charged defect concentrations and thus the diffusivity [111]. The intrinsic carrier density of Si and Ge at T = 890 oC are 3.18 × 1018 cm-3 and 1.37 × 1019 cm-3 respectively, which are below the As/P doping concentration.  This indicates that the Fermi-level effect existed during the annealing.  In our group, Cai et al. has investigated Si-Ge interdiffusion with a high P doping level by both experiments and modeling in the range of 0.75 < 𝑥𝐺𝑒  < 1 [108]. The doping dependence of Si-Ge interdiffusion was modeled successfully by a Fermi-enhancement factor, which will be elaborated in the following text. 96  It has been reported in the literature that Si diffusion in Ge and Ge diffusion in Si are both mediated mostly by vacancies [88][168], and Si–Ge interdiffusion with P/As doping is mediated by negative point vacancies [108]. For Si-Ge interdiffusion, the total interdiffusion coefficient is given by the sum of the individual contributions of 𝐷𝑆𝑖,𝐺𝑒𝑉𝑟− , i.e.: 𝐷𝑡𝑜𝑡𝑎𝑙 = ∑ 𝐷𝑆𝑖,𝐺𝑒𝑉𝑟−2𝑟=0 =1𝐶0∑ 𝑓𝑟𝐶𝑉𝑟−𝑒𝑞 ?̃?𝑉𝑟−2𝑟=0                                                        (4.6),                                                                  where C0 is the atom density of 𝑆𝑖1−𝑥𝐺𝑒𝑥 ; 𝑓𝑟  is the diffusion correlation factor. For diffusion via vacancies in diamond structures like in 𝑆𝑖1−𝑥𝐺𝑒𝑥 , 𝑓𝑟  is considered to be independent of the charge state and is set to be 𝑓𝑟 = 𝑓𝑉 = 0.5 [113]; 𝐶𝑉𝑟−𝑒𝑞  is the thermal equilibrium concentration of 𝑉𝑟−  point defects, and 𝐷𝑉𝑟−  is the diffusion coefficient mediated by 𝑉𝑟− with 𝑟 ∈ {0,1,2} , respectively. Cai et al. [108] found that the interdiffusion is mainly dominated by 𝑉2−  point defects, which means 𝐷𝑉1− = 𝑚1𝐷𝑉0  and 𝐷𝑉2− = 𝑚2𝐷𝑉0 , where 𝑚1 = 1, 𝑚2  >= 20. Therefore, Equation (4.6) can be transformed to Equation (4.7):  D̃𝑡𝑜𝑡𝑎𝑙 = D̃𝑉0𝐶𝑉0𝑒𝑞𝐶0𝑓𝑟 (1 + 𝑚1𝐶𝑉−𝑒𝑞𝐶𝑉0𝑒𝑞 + 𝑚2𝐶𝑉2−𝑒𝑞𝐶𝑉0𝑒𝑞 )  (𝑚1 = 1, 𝑚2 ≥ 20)                (4.7).  𝐶𝑉−𝑒𝑞 = 𝐶𝑉0𝑒𝑞𝑒𝑥𝑝 (𝐸𝐹−𝐸𝑉−𝑘𝑇)                                                                   (4.8) 𝐶𝑉=𝑒𝑞 = 𝐶𝑉0𝑒𝑞𝑒𝑥𝑝 (2𝐸𝐹 −𝐸𝑉−−𝐸𝑉=𝑘𝑇)                                                          (4.9)       Charged point defects have energy levels in the bandgap, and the occupation of defect related energy levels depends on the position of the Fermi level Ef, which is a function of the dopant concentration. If the doping concentration exceeds the intrinsic carrier concentration ni, the Fermi level Ef will deviate from its intrinsic position Ei. The ratio of 97  the charged vacancy concentration to the neutral vacancy concentration is given by Equations (4.8) and (4.9). Combining Equation (4.7) with Equation (4.8), (4.9) and (2.6), the total interdiffusivity can be expressed as: 𝐷𝑡𝑜𝑡𝑎𝑙 (𝑛) = D̃𝑉0𝐶𝑉0𝑒𝑞𝐶0𝑓𝑟 [1 + ∑ (𝑛𝑛𝑖)𝑟𝑚𝑟𝑒𝑥𝑝 (𝑟𝐸𝑖 −∑ 𝐸𝑉𝑛−𝑟𝑛=1𝑘𝑇)2𝑟=1 ]               (4.10), when n = ni, the total diffusion coefficient is equal to the intrinsic interdiffusivity, 𝐷(𝑛𝑖). Therefore, the ratio between extrinsic and intrinsic diffusion coefficient is: ?̃?(𝑛)?̃?(𝑛𝑖)=1+∑ (𝑛𝑛𝑖)𝑟𝑚𝑟𝑒𝑥𝑝 (𝑟𝐸𝑖−∑ 𝐸𝑉𝑛 −𝑟𝑛=1𝑘𝑇)2𝑟=11+∑ 𝑚𝑟𝑒𝑥𝑝 (𝑟𝐸𝑖−∑ 𝐸𝑉𝑛−𝑟𝑛=1𝑘𝑇)2𝑟=1  ( 𝑚1 = 1, 𝑚2 ≥ 20 ) .        ( 4 . 1 1 ) For the energy term 𝑟𝐸𝑖 − ∑ 𝐸𝑉𝑛−𝑟𝑛=1  (𝑟 ∈ {1, 2}) , they are functions of 𝑥𝐺𝑒  in𝑆𝑖1−𝑥𝐺𝑒𝑥. However, due to limited literature resources of the energy levels of 𝑉− and 𝑉2− in SiGe, i.e. 𝐸𝑉− (𝑥𝐺𝑒), andEV2−(𝑥𝐺𝑒), this terms was linearly interpolated between the value in Si and Ge, i.e.: 𝐴𝑟,𝑆𝑖𝐺𝑒(𝑥𝐺𝑒) = 𝐴𝑟,𝑆𝑖 (1 − 𝑥𝐺𝑒) + 𝐴𝑟,𝐺𝑒𝑥𝐺𝑒 ,    (0 < 𝑥𝐺𝑒 < 1),                       (4.12) where 𝐴𝑟,𝑆𝑖𝐺𝑒  refers to the energy term 𝑟𝐸𝑖 − ∑ 𝐸𝑉𝑛−𝑟𝑛=1  (𝑟 ∈ {1, 2})  in SiGe. For Si, 𝐴1,𝑆𝑖 = 0.1383 eV  and 𝐴2,𝑆𝑖 = −0.1835 eV  [111]. For Ge, 𝐴1,𝐺𝑒 = −0.1134 eV  and 𝐴2,𝐺𝑒 = 0.0866 eV. [169]  As for the calculation of electron density n, considering the charge neutrality equation 𝑛 = 𝑝 + 𝐶𝑃/𝐴𝑠  and 𝑛𝑖2 = 𝑛𝑝 , the electron concentration n of the P-doped 𝑆𝑖1−𝑥𝐺𝑒𝑥 samples can be expressed as: 𝑛(𝑥𝐺𝑒 , 𝐶𝑃/𝐴𝑠 ) =𝐶𝑃/𝐴𝑠+√𝐶𝑃/𝐴𝑠2+4𝑛𝑖2(𝑥𝐺𝑒)2 .                                                (4.13) 98  The last parameter is the calculation of 𝑛𝑖 (𝑥𝐺𝑒). Due to the limited data of 𝑛𝑖 (𝑥𝐺𝑒) at temperature over 600 oC, Cai et al. used linear interpolation between 𝑛𝑖,𝐺𝑒  and 𝑛𝑖,𝑆𝑖  in his original code as Equation (4.14). This approximation is good enough over the range 0.75 < 𝑥𝐺𝑒  < 1. However, our work covers 0 < 𝑥𝐺𝑒  < 1 range. The method used by Cai et al. overestimates 𝑛𝑖 (𝑥𝐺𝑒) when 𝑥𝐺𝑒  < 0.75. Since at 𝑥𝐺𝑒  < 0.85, 𝑆𝑖1−𝑥𝐺𝑒𝑥  alloys have been always considered as “Si-like” material due to its band structure and electronic properties [166]. Thus, we also tried exponential interpolation between 𝑛𝑖,𝐺𝑒  and 𝑛𝑖,𝑆𝑖 as Equation (4.15).  𝑛𝑖 (𝑥𝐺𝑒) = 𝑛𝑖,𝐺𝑒𝑥𝐺𝑒 + 𝑛𝑖,𝑆𝑖 (1 − 𝑥𝐺𝑒)                                                          (4.14) 𝑛𝑖 (𝑥𝐺𝑒) = 𝑛𝑖,𝑆𝑖 exp (𝑙𝑛𝑛𝑖,𝐺𝑒𝑛𝑖,𝑆𝑖× 𝑥𝐺𝑒)                             ( 4 . 1 5 ) The two 𝑛𝑖(𝑥𝐺𝑒) models are illustrated in Figure 4.11. The simulation results with both approximation methods will be shown in Section 4.2.6.    Figure 4.11 Comparison between the calculation results of the two intrinsic carrier density model at T = 890 oC.  99  4.2.6 Simulation of Ge profiles after annealing  Since both sample LB-5TC and HB-5TC have similar Ge profiles with sample U-5TC, it means B has little impact on Si-Ge interdiffusion. We will focus our simulation work on sample A-5TC and P-5TC. According to Equation (4.11) and (4.13), the extrinsic interdiffusion coefficient increases with the concentration of dopant (P/As). However, as shown in Figure 4.6, during annealing, the concentration profile of dopant changed due to dopant diffusion and segregation. Ideally, it is best to simulate dopant diffusion, dopant segregation, and Si-Ge interdiffusion simultaneously. However, the diffusion and segregation of dopant involve many unknown coefficients such as the P/As diffusion segregation coefficients as a function of 𝑥𝐺𝑒, which are beyond the scope of this study. Therefore, in this simulation study, as discussed in Section 4.2.1, an approximation method was used that treated P and As concentration profile as unchanged during the process of Si-Ge interdiffusion. We took the annealed P/As concentration profiles as our background dopants profiles.  The simulation was done by Matlab to calculate 𝐷(𝑛) in Equation (4.10) and to simulate interdiffusion profiles using Fick’s second law: 𝜕𝐶𝐺𝑒𝜕𝑡=𝜕𝜕𝑧(𝐷(𝑛)𝜕𝐶𝐺𝑒𝜕𝑧).                                                                                          (4.16) To solve the diffusion equation numerically, we used finite difference time domain (FDTD) method. Details of this method are included in Appendix A. Numerically, we treated the sample as 1D material and divided up the samples into planes with different Ge concentration. The distance between each plane ∆x is 1nm. The total annealing time t is divided into small time period of ∆t (0.001 second).  Smaller ∆x and ∆t have also been 100  tested to verify the reliability of ∆x = 1nm and ∆t = 0.001s. The results showed they are small enough.  The experimental data of A-NoTC and P-NoTC were used as the initial profiles of the simulation. The boundaries are chosen to be far away from the interdiffusion region according to experimental data.  As for the simulation temperature, as elaborated in Section 4.2.4, we believe the extracted annealing temperate should be 890 oC and 870 oC for Sample P-5TC and A-5TC separately. We also calculated simulation at T = 850 oC for Sample P-5TC and at T = 830 oC for Sample A-5TC for comparison. The results are shown in Figure 4.12. Another critical parameter in simulation is 𝑛𝑖 (𝑥𝐺𝑒). Due to the limited data, two different models have been adopted. One is the exponential model based on Equation (4.15), and the other is the linear model based on Equation (4.14).   As illustrated in Figure 4.12, using T=890/870 oC and 𝑛𝑖 (𝑥𝐺𝑒) exponential model can give out the best Ge profiles fitting results for sample P-5TC/A-5TC respectively. According to Figure 4.12(a), although P-Simu_Ni=linear_T=890 has a better fitting curve with P-5TC in 𝑥𝐺𝑒  > 0.6 part compared with P-Simu_Ni=expo_T=890, it has a plateau in 0.4 < 𝑥𝐺𝑒  < 0.6. This plateau is not real, but a result due to the underestimation of interdiffusion enhancement in 𝑥𝐺𝑒  > 0.6 region. Using 𝑛𝑖 (𝑥𝐺𝑒) exponential model can solve this problem. Thus, we conclude that 𝑛𝑖 (𝑥𝐺𝑒)  with an exponential dependence on 𝑥𝐺𝑒  is a better model than the 𝑛𝑖 (𝑥𝐺𝑒)  with a linear dependence on 𝑥𝐺𝑒 . The extracted temperature (890/870 oC) also turns out to work better than the nominal reading temperature (850/830 oC) for Sample P-5TC/A-5TC.  101   At 𝑥𝐺𝑒  > 0.5 part, P-Simu_Ni=expo_T=890 and P-5TC Ge profiles are not perfectly matched. That inconsistence should be due to the over simplified treatment of P profile. According to Kirkendall effect, as Si diffuses much faster in Ge than Ge in Si, lattice planes will be created in Ge sides and be destroyed in Si sides. These creation and annihilation of lattice planes might result in the shift of P profile towards the Si side during the annealing procedure. That means we would underestimate the Fermi-enhancement effect at the Ge side if we treated P profile as unchanged. For sample A-5TC simulation results, we don’t have similar problem. It should be due to the lower concentration of As and lower annealing temperature, making Kirkendall effect less significant.  Figure 4.7 shows P profile movement at 870 oC from 50 minutes annealing to 200 minutes annealing. P has a segregation peak at the Ge/Si interface due to high density of defects and dislocations. The lateral movement of P profile to the Si side is 50 - 60 nm. This is larger than the possible error of ± 50 nm, which is from the epitaxial growth of non-uniformity of ± 30 nm and the SIMS depth error of ± 20 nm. Therefore, Kirkendall effect did lead to the lateral shifting of P during the annealing. The P profiles can be used as an indicator of the Ge/Si.  102   Figure 4.12 Simulation results with different parameters (a) sample P-5TC; (b) sample A-5TC. Temperature is set to be 890/850 oC for sample P and 870/850 oC for sample A. The 𝐧𝐢(𝒙𝑮𝒆) is set to change exponentially or linearly with 𝒙𝑮𝒆. 103    4.3 Chapter Summary In this chapter, we first presented the Ge and dopant profiles of samples before and after annealing. It shows that P and As enhance the interdiffusion of Si-Ge significantly while B has little impact. the Boltzmann-Matano method has been adopted to extract the time-averaged Si-Ge interdiffusivity as a function of 𝑥𝐺𝑒 . By comparing sample U-5TC and other well-established models, the extracted annealing temperatures of sample A-5TC and sample P-5TC have been determined. Attributing the enhancement of interdiffusion to Fermi-level effect, we successfully simulated the Ge profiles of sample P-5TC and A-5TC. The doping dependence of the Si-Ge interdiffusion can be described by the following equation: ?̃?(𝑛)?̃?(𝑛𝑖)=1+∑ (𝑛𝑛𝑖)𝑟𝑚𝑟𝑒𝑥𝑝 (𝑟𝐸𝑖−∑ 𝐸𝑉𝑛 −𝑟𝑛=1𝑘𝑇)2𝑟=11+∑ 𝑚𝑟𝑒𝑥𝑝 (𝑟𝐸𝑖−∑ 𝐸𝑉𝑛−𝑟𝑛=1𝑘𝑇)2𝑟=1 (𝑚1 = 1, 𝑚2 ≥ 20)  .          (4.17) Due to the limited data, the intrinsic carrier density 𝑛𝑖(𝑥𝐺𝑒)  was assumed to change linearly or exponentially with 𝑥𝐺𝑒. The simulation results proved that exponential model predicts the interdiffusion profile more accurately.     104  Chapter 5: Summary, Contributions and Future Work 5.1 Thesis Summary Ge, as the most Si-compatible semiconductor, plays an important role both in Si photonics and in SiGe-based electronic devices such as MOSFETs and HBTs. The Recent development of Ge-on-Si lasers and III-V lasers on Ge/Si requires more understanding of Ge film quality and the interdiffusion at Ge/Si interface such as the doping impact. This work studied Ge-on-Si growth and Si-Ge interdiffusion with different doping conditions both by experiments and theoretical modeling. Below is the summary of this thesis work. 1. We designed the Ge-on-Si structures with five (U, P, A, LB and HB) different doping configurations and with two annealing conditions (5TC and HT). The two annealing conditions had no significant difference since they had the same high-temperature annealing time (850 oC for 50 minutes). By AFM and HRXRD analysis, we found that different types of doping had no significant impact on the surface roughness and strain degree. All samples have a smooth surface (roughness < 2 nm). The Ge films are almost entirely relaxed. On the other hand, EPD, AFM and TEM results showed that B introduces lots of extra TDs (TDD > 1 × 108 cm-2) in the bulk Ge layer compared to the undoped samples. This greatly impairs the Ge films quality. The TDD value of undoped sample without annealing (U-No TC) is 3.5 ± 1.5 × 106 cm-2. Although the annealing procedure can reduce the TDD to less than 105 cm-2, it is compromised by interdiffusion between Si and Ge layers, which is highly undesired for device performance. Meanwhile, P and As doping can reduce the TDD values of 105  unannealed samples (P-No TC and A-No TC) to low 105 cm-2. This offers a new method to fabricate high-quality Ge-on-Si films without defect annealing procedure, which is one of the major findings of this work.  2. SIMS measurement results showed that Si-Ge interdiffusion is greatly enhanced in P and As doped samples, while B doping had little impact on that. After ruling out the possibility of strain and dislocation impact, we attributed this phenomenon to Fermi-level effect from P or As doping, which increases the negatively charged vacancy concentrations and thus the interdiffusivity. We used the extrinsic n-doped Si-Ge interdiffusion model with the Fermi-enhance factor to describe the impact of the Fermi-level effect. For the intrinsic carrier density as a function of 𝑥𝐺𝑒, we used an exponential model instead of an linear model. The validity of the model was proved by the comparisons between the simulations and the SIMS data from experiments with different anneal temperatures in 0 < 𝑥𝐺𝑒 < 1  range. It was found that Si-Ge interdiffusivity is proportional to 𝑛2 /𝑛𝑖2. This result suggested that Si-Ge interdiffusion is dominated by V2- point defects across all 𝑥𝐺𝑒.  5.2 Contributions and Technological Implication The main contributions of this thesis work are: 1) we discovered that the use of P and As doping during epitaxial growth can greatly enhance the Ge film quality without defect annealing, and 2) we refined the previous models on Si-Ge interdiffusion with n-106  type doping. The refined model is more accurate than the previous one, and the valid range is expanded to the full Ge concentration range. The results of this thesis have an impact on the design of optical and electronic devices with Ge-on-Si structures, including III-V lasers on Ge/Si, Ge-on-Si lasers, Ge modulators, and Ge photodetectors. Since P and As can reduce the TDD in the Ge layers, we can use this method to fabricate high-quality Ge-on-Si films without defect annealing procedure, especially for the devices require n-type doping and minimal Si-Ge interdiffusion, like Ge-on-Si lasers. For the devices require p-type doping, like optical modulators and avalanche photodetectors, using alternative dopants instead of boron can be a better option since boron will severely impair the Ge film quality.  Given that high concentration of P/As accelerate Si-Ge interdiffusion drastically, for a laser with a Ge layer thickness of several hundreds of nanometers, an interdiffused region can change a significant portion of the active Ge layer back to SiGe alloys. Thus, it will greatly offset the effort in making Ge into a direct bandgap material. It is important to estimate the interdiffusion range in such a device, especially when it ’s n-type doped, like Ge-on-Si lasers. We can use the interdiffusion model developed in this work to predict that.  5.3 Future Work In this thesis, the Ge-on-Si films quality under P, As and B doping has been investigated. The Si-Ge interdiffusion under P and As highly doping conditions have been successfully modeled. This model can be used to predict n-type doped extrinsic Si-Ge 107  interdiffusion across all 𝑥𝐺𝑒. However, due to the scope of this thesis, many aspects of extrinsic Si-Ge interdiffusion are not addressed in this work, and can be topics for future studies: Topic 1: Simulate P/As diffusion and Si-Ge interdiffusion simultaneously.  In this work, P and As diffusion in Chapter 4 was neglected to simplify the simulation. However, in reality, we expect P/As profile shall laterally shift towards Si direction due to Kirkendall effect as discussed in Section 4.2.1. From Figure 4.7, we can see that that the P profile movement from 50 minutes annealing to that in 200 minutes annealing is 50-60 nm. That speed is pretty slow but not negligible. It can be critical in certain kinds of optical and electronic devices, such as Ge-on-Si laser with asymmetric P doping concentration in Ge and Si side. For better prediction, it is highly recommended to build a multi-element (P/As, Si, and Ge) diffusion model to simulate the n-type doped Si-Ge system. More experimental data is required to analysis that quantitatively. Topic 2: Investigate the interactions between impurities and dislocations in Ge-on-Si films.  According to the EPD results in Section 3.2, P/As can reduce TDD values in samples without annealing. On the other hand, the P/As doped samples after annealing have higher TDD values compared with the undoped sample. B will introduce lots of extra TDs with or without annealing. The mechanism behind it remains unclear. Topic 3: Impact of the n-type doping on the photoluminescence (PL) and lasing of Ge. 108  This work has used many material characterization techniques to investigate the Ge film quality. For Ge-on-Si lasers, it is important to measure the photoluminescence directly to find a better growth and processing condition for lasing. Due to the lack of a PL set up with low temperature and Ge laser fabrication capability, we were not able to perform these tasks. We have reached out to Prof. Jifeng Liu from Dartmouth College, who pioneered Ge lasers. His group has started the PL measurement work and analysis.  Although Ge and Si based microelectronics are not discussed in this thesis, the knowledge gained here will also help the development of Ge-based MOSFETs. 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Scilla, “Experimental evidence of both interstitial‐ and vacancy‐assisted diffusion of Ge in Si,” Appl. Phys. Lett., vol. 54, no. 9, pp. 843–845, Feb. 1989.   120  Appendices Appendix A: Overview of the Simulation Implement in MatlabTM In this work, we used MatlabTM to simulate Si-Ge interdiffusion in SiGe heterostructure. To solve the diffusion equation numerically, finite difference time domain method was used. Numerically, we divided up the samples into planes ∆𝑥 apart, labeled 0, 1, 2 …, and at each plane, the concentration  of atoms is 𝐶0, 𝐶1, 𝐶2 … . For the annealing time, the total time t was divided into N periods with ∆𝑡 for each period. Therefore at plane i, the analytical equation of Fick’s second law can be written numerically as: 𝐶𝑖+−𝐶𝑖∆𝑡= 𝐷𝑖𝐶𝑖+1−𝐶𝑖 +𝐶𝑖 −1−𝐶𝑖∆𝑥2,                                                                                           (A1) where 𝐷𝑖 is the diffusion (interdiffusion) coefficient in each plane and 𝐶𝑖+  is the atom concentration at plane i  after time period ∆𝑡 . Therefore, after each period ∆𝑡 , the concentration of atoms in each plane can be expressed as: 𝐶𝑖+ = 𝐶𝑖 +𝐷𝑖 ∆𝑡∆𝑥2(𝐶𝑖+1 − 2𝐶𝑖 + 𝐶𝑖−1).                                                                         (A2) Given the initial concentrations at different places and boundary conditions, we can calculate the subsequent concentration after time t. However, parameters ∆𝑡 and ∆𝑥 should be chosen such that they are relevant to the problem we are attempting to solve. For example, ∆𝑥 should be such that the profile is uniformly divided up into a sufficient number of distance intervals that the profile can be fitted in a reasonable manner by these piecewise linear approximations. Similarly, ∆𝑡 should be chosen so that the time interval is divided into a sufficient number of time steps to resolve the diffusion process .  In 121  addition, parameters, ∆𝑥 and ∆𝑡, should satisfy  𝐷∆𝑡∆𝑥2≤12 such that 𝐶𝑖+will always larger than 0.  The codes of the modeling in MatlabTM are shown as follows:         %This program is used to simulate the sample P-5TC using Fermi-level effect %the author is Guangnan Zhou   clc; close all; clear all;   %% intial settings k=8.6173324e-5;     %boltzmann constant T=273.15+890;       %annealing temperature     time=50*60;   dt=0.01;      %annealing time and time step dx=1;  depth=50:dx:850; N=length(depth);       %% load data and smooth load P_5TC_profile.txt          %depth in nm, Ge in fraction, P in cm^-3  load P_NoTC_Ge_profile.txt      %depth in nm, Ge in faction load       P_5TC_profile(:,1)=P_5TC_profile(:,1); P_anneal_depth=P_5TC_profile(:,3); P_anneal_raw=P_5TC_profile(:,4); y=smooth(P_anneal_raw,5); P_anneal=spline(P_anneal_depth,y,depth); Xge_anneal=spline(P_5TC_profile(:,1),P_5TC_profile(:,2),depth);   rawdata=P_NoTC_Ge_profile(:,2); y=smooth(rawdata,10); Xge_asgrown=spline(P_NoTC_Ge_profile(:,1),y,depth);   % figure % subplot(2,1,1); % plot(P_NoTC_Ge_profile(:,1),rawdata,depth,Xge_asgrown) % title('P-NoTC before and after smooth') % subplot(2,1,2); % semilogy(P_anneal_depth,P_anneal_raw,depth,P_anneal) % legend('before','after') % title('P profile in P-5TC before and after smooth')    %% 122  Cge_anneal = (25-(5-Xge_anneal*1.16).^2)*10/2.32;      %concentration in nm^-3 Cge_asgrown = (25-(5-Xge_asgrown*1.16).^2)*10/2.32;    %concentration in nm^-3 boundaryshift=(Cge_asgrown(1)-Cge_anneal(1))/time;     %the boundary of asgrown and anneal are different Cge = Cge_asgrown; const = dt/dx^2; step=0;   for i=1:dt:time     Xge=(5-sqrt(25-2.32*Cge/10))/1.16;     if any(Xge<0)         disp('Xge<0')         return     elseif any(Xge>1.01)         disp('Xge>1.01')         return end      D=Dfunc(Xge,P_anneal,T,N);     Cge2=Cge;                  %Cge2 as the next moment concentration         for j=2:N-1         Cge2(j)=Cge(j)+D(j-1)*const*(Cge(j-1)-Cge(j))+D(j)*const*(Cge(j+1)-Cge(j));         %Cge(j)=Cge(j)+D(j)*const*(Cge(j-1)-2*Cge(j)+Cge(j+1));     end          Cge=Cge2;     step=step+1;           end   figure plot(depth,Cge_asgrown,depth,Cge,depth,Cge_anneal,'LineWidth',2) legend('as grown','simulation results','annealed') xlabel('depth/nm','FontSize',16);ylabel('X_G_e','FontSize',16); title(sprintf('depth VS Xge (xshift=%d)',xshift),'FontSize',20) set(gca,'FontSize',16);    function D=Dfunc(x1,p1,T,N)   x=zeros(1,N-1); p=zeros(1,N-1);   for i1=1:N-1     x(1,i1)=(x1(1,i1+1)+x1(1,i1))/2;     p(1,i1)=(p1(1,i1+1)+p1(1,i1))/2; end   123  k=8.6173324e-5; DU=0.01*exp(2.5*x)*exp(-3.1/k/T)+1.12*10^(-4)*exp(12.75*x)*exp(-3.33/k/T); D_lat=DYW(x,T); D_dis=DU-D_lat;   NiGe=9.7E15*T^1.5*exp(-4350/T);NiSi=2.8E16*T^1.5*exp(-6450/T); ni=x*NiGe+(1-x)*NiSi; n=(p+sqrt(p.^2+4*ni.^2))/2; n_ratio=n./ni;   a1 = -0.2516; b1 = 0.13825; m1 = 1; a2 = 0.0969; b2 = -0.1835; m2 = 100; beta=m1*exp((a1*x+b1)/k/T);gamma=m2*exp((a2*x+b2)/k/T); f_factor = (1+beta.*n_ratio+gamma.*n_ratio.^2)./(1+beta+gamma);   D=D_dis+D_lat.*f_factor; D=D*1e14;  %%the function that use DYW's model function Dtot=DYW(x,T)   k=8.6173324e-5; Rg=8.314472;   EaGe=4.83*(1-x)+3.13*x+1.63*x.*(1-x); EaSi=4.76*(1-x)+3.32*x+1.54*x.*(1-x); D0Si=exp(6.849+4.964*x-7.829*x.^2); D0Ge=exp(6.636+8.028*x-11.318*x.^2); DSi=D0Si.*exp(-EaSi/(k*T)); DGe=D0Ge.*exp(-EaGe/(k*T));   dlnSi=(1-x).*x.*(4017*x-17574)/(Rg*T); dlnGe=(1-x).*x.*(4017*x-18913)/(Rg*T); Dtot=DSi.*(1+dlnSi).*x+DGe.*(1+dlnGe).*(1-x); end     124  Appendix B: Strain Simulation with software PANalytical Epitaxy PANalytical Epitaxy is a part of PANalytical’s software package, which provides functionality for plotting and analyzing high-resolution X-ray diffraction data of rocking curves, 2-axes scans, reciprocal space maps and wafer maps. Epitaxy offers a wealth of key information on thin heteroepitaxial layers, particularly single-crystal and highly textured thin-layer samples, such as lattice mismatch and relaxation, composition and layer thickness. In PANalytical Epitaxy, the relaxation R is used to model the change in the unit cell distortion that occurs in the imperfect interfaces where misfit dislocations are present. This parameter is used to calculate the correct peak positions in simulated rocking curves. The expression of the percentage relaxation can be written as: 𝑅𝑒𝑙𝑎𝑥𝑎𝑡𝑖𝑜𝑛 =  (𝑎𝐴∗ −𝑎𝐵∗ )(𝑎𝐴−𝑎𝐵∗ )× 100%,                                                                        (B1) where 𝑎𝐴∗  and 𝑎𝐵∗  are the in-plane lattice parameters of layer A and layer B with partial strain relaxation, and 𝑎𝐴  is the in-plane lattice parameter of layer A with no strain. For the  Figure B1 A typical scenario of mismatch strain between two layers with a cubic crystalline structure. The lattice parameter of B is larger than that of A, 𝒂𝑩 > 𝒂𝑨. Layer ALayer B125  fully strained case, i.e. 𝑎𝐴∗ = 𝑎𝐵∗ , the relaxation R is 0%; for the fully relaxed case, i.e. 𝑎𝐴∗ =𝑎𝐴 , the relaxation R is 100%.   The strain level in layer A can be expressed as:  𝜀𝐴 =𝑎𝐴∗𝑎𝐴× 100% − 1,                                                                                         (B2) By combining Ea. (B1) and (B2), we can obtain the strain level in layer A as: 𝜀𝐴 =𝑅(𝑎𝐴−𝑎𝐵∗ )+𝑎𝐵∗𝑎𝐴-1,                                                                                             (B3) where 𝑎𝐴  and 𝑎𝐵∗  can be found from literature or previous calculation. In this work, PANalytical Epitaxy is employed to do the rocking curve analysis for samples in Chapter 3 and Chapter 4. The information of the strain relaxation in the epitaxial layers can be derived from the XRD peak positions. After achieving the relaxation R of the epitaxial layer, we can calculate the strain level of target epitaxial layer by using Ea. (B3).  The simulation procedure is shown as follows. For the simulations in PANalytical Epitaxy, firstly, an identical multi-layered structure as the real sample is built, shown in Figure B2. The peak position of target layer in simulated XRD profile is determined by two parameters (R and Ge fraction). For the value of Ge fraction, we used the data that  obtained from SIMS analysis. By changing the input value of relaxation R of Ge layer, the simulation peaks can be matched with the experimental peak positions (relative to the substrate peak), as shown in Figure B3. More instructions are described in the guide of the sotware PANalytical Epitaxy. 126       Figure B2 An identical multi-layered structure built in PANalytical Epitaxy simulation.  Figure B3 Peak matching in PANalytic Epitaxy for sample U-No TC in Chapter 3 with R = 104% simulation. 

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