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PAGE 1 1 CHARACTERIZATION AND MODELING OF STRAINED SI FET AND GAN HEMT DEVICES By MIN CHU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 201 1 PAGE 2 2 201 1 Min Chu PAGE 3 3 To my family PAGE 4 4 ACKNOWLEDGMENTS I sincerely thank my advisor, Prof Scott E. Thompson, for his support, guidance, and giving me the opportunity of entering the challenge field of device characterization, analysis and design advis or, Prof Toshikazu Nishida, for his guidance, encouragement and sharing of knowledge. Also, I thank my committee members, Prof Jing Guo and Prof Bhavani Sankar, for participating and evaluating my research work. And I thank Prof. Susan Sinnott for her instruction and discussion on the DFT topic. I would like to especially thank all my colleagues : Amit, Andy, Guangyu, Hyunwoo, Jingjing, Ji S ong, Kehuey, Lu Minki, Nidhi, Onur, Sagar, Sri, Tony, Toshi, Ukjin, Uma, Xiaodong, Yongke, Younsung for their help and discussion during my graduate study Finally, I would like to show my greatest appreciation to my parents and my husband for their endless support. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURES ................................ ................................ ................................ .......... 9 LIST OF ABBREVIATI ONS ................................ ................................ ........................... 12 ABSTRACT ................................ ................................ ................................ ................... 13 CHAPTER 1 STRAIN ENGINEERING IN STATE OF THE ART SI FET AND GAN HEMT ........ 15 1.1 Motivation ................................ ................................ ................................ ......... 15 1.2 Brief History of Strained Semiconductors ................................ ......................... 16 1.2.1 Strained Si FET ................................ ................................ ....................... 16 1.2.2 Strained GaN HEMT ................................ ................................ ................ 19 1.3 Objectives and Organization ................................ ................................ ............. 20 2 FUNDAMENTAL PHYSICS BEHIND STRAINED SI AND GAN DEVICES ............ 22 2.1 Strain and Stress ................................ ................................ .............................. 22 2.1.1 Stra in Definition ................................ ................................ ....................... 22 2.1.2 Piezoresistance Coefficients ................................ ................................ ... 23 2.1.3 Wafer Bending ................................ ................................ ......................... 24 2.2 Strained Si D evices ................................ ................................ ........................... 27 2.2.1 Stress Effects on N Si Band Structure ................................ ..................... 28 2.2.2 Stress Effects on P Si Band Structure ................................ ..................... 31 2.3 Strained G aN D evices ................................ ................................ ...................... 33 3 PIEZORESISTANCE OF SI DEVICES ................................ ................................ ... 38 3.1 Introduction ................................ ................................ ................................ ....... 38 3.2 Piezoresista nce of Planar n and pMOSFETS ................................ .................. 40 3.2.1 Device and Experiment Details ................................ ............................... 40 3.2.2 Results and Discussion ................................ ................................ ........... 41 3.2. 2.1 Stress results in n type devices ................................ ..................... 41 3.2.2.2 Stress results in p type devices ................................ ..................... 45 3.3 Piezoresistance of TG FinFETs ................................ ................................ ........ 48 3.3.1 Device and Experiment Details ................................ ............................... 48 3.3.2 Experiment al Results ................................ ................................ ............... 49 3.3.3 Discussion ................................ ................................ ............................... 52 3.3.3.1 A model for strain enhanced TG FinFETs ................................ ...... 53 PAGE 6 6 3.3.3.2 N channel TG FinFET behavior ................................ ..................... 54 3.3.3.3 P channel TG FinFET behavior ................................ ..................... 54 3.4 Conclusion ................................ ................................ ................................ ........ 55 4 EFFECT OF STRESS ON GAN HEMT RESISTANCE ................................ ........... 57 4.1 Introduction ................................ ................................ ................................ ....... 57 4.2 Theory and Modeling ................................ ................................ ........................ 57 4.2.1 Stress Dependence of 2DEG Sheet Carrier Density ............................... 58 4.2.2 Stress Dependence of Channel Electron Mobility ................................ ... 60 4.2.3 Simulation Uncertainty ................................ ................................ ............. 64 4.3 Results and Discussion ................................ ................................ ..................... 65 4.4 Conclusion ................................ ................................ ................................ ........ 67 5 EFFECT OF STRESS ON G A N HEMT GATE LEAKAGE ................................ ...... 72 5.1 Motivation ................................ ................................ ................................ ......... 72 5.2 Gate Leakage Mechanisms ................................ ................................ .............. 73 5.2.1 Literature Review ................................ ................................ ..................... 74 5.2.2 Di rect Tunneling ................................ ................................ ...................... 75 5.2.3 Bulk Trap Assisted Leakage ................................ ................................ .... 78 5.2.4 Poole Frenkel Emission from Surface States ................................ .......... 81 5.3 Effects of Stress on Gate Leakage ................................ ................................ ... 85 5.3.1 Stress Dependent Parameters ................................ ................................ 85 5.3.2 Results and Discussions ................................ ................................ ......... 89 5.4 Conclusion ................................ ................................ ................................ ........ 89 6 DFT CALCULATION FOR GAN ................................ ................................ ............. 92 6.1 DFT Introduction ................................ ................................ ............................... 92 6.1.1 Basic Concept of DFT ................................ ................................ ............. 92 6.1.2 Why Choose DFT ................................ ................................ .................... 95 6.1.3 DFT Ca lculation Procedure Using VASP ................................ ................. 96 6.2 DFT Calculation for Bulk GaN ................................ ................................ ........... 99 6.2.1 Standard DFT Calculation ................................ ................................ ....... 99 6.2.2 Bandgap Correction ................................ ................................ ............... 101 6.3 Ga N Defect Calculations ................................ ................................ ................. 104 6.3.1 Literature Review ................................ ................................ ................... 104 6.3.2 Test Calculation and Computational Issues ................................ .......... 105 6.4 Conclusion ................................ ................................ ................................ ...... 106 7 SUMMARY AND RECOMMENDATIONS FOR FUTURE WORK ......................... 107 7.1 Summary ................................ ................................ ................................ ........ 107 7.2 Recommendations for Future Work ................................ ................................ 108 APPENDIX PAGE 7 7 A INCAR FILES ................................ ................................ ................................ ........ 110 B POSCAR FILES ................................ ................................ ................................ .... 112 C KPOINTS FILES ................................ ................................ ................................ ... 114 LIST OF REFERENC ES ................................ ................................ ............................. 115 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 130 PAGE 8 8 LIST OF TABLES Table page 3 1 Experimental extracted coefficients of Si planar nMOSFETs.. ........................ 43 3 2 Experimental extracted coefficients of Si planar pMOSFETs ......................... 46 3 3 The longitudinal coefficients ( 10 12 dyne/cm 2 ) for n and p channel TG FinFETs ................................ ................................ ................................ ............ 50 4 1 Gauge factors of GaN HEMT s and bulk GaN published in literature. ................. 71 4 2 The best fit set of stiffness constants of GaN and AlN ................................ ...... 71 4 3 The best fit set of piezoelectric coefficients of GaN and AlN ............................ 71 5 1 The Poole Frenkel Emission fitting parameters. ................................ ................. 85 6 1 The standard DFT PBE calculation results for the GaN 4 atom unit cell and 32 atom supercell structures, with and without external stress. ........................ 100 6 2 GaN bandgaps calculated from the DFT PBE, LDA+U, and HSE functionals. 103 A 1 The INCAR file for the self consistent (SC) calculation step in the standard DFT model. ................................ ................................ ................................ ....... 110 A 2 List of the modified settings in the INCAR file for the band structure calculation step in the standard DFT model. ................................ .................... 111 A 3 List of INCAR flags that defines the +U correction in this work. ........................ 111 A 4 List of INCAR flags that defines the HSE calculation in this work. .................... 111 PAGE 9 9 LIST OF FIGURES Figure page 1 1 Illustration of process induced stress on Si MOSFETs. ................................ ...... 18 2 1 Illustrations of mechanically bended wafer samples. ................................ .......... 25 2 2 Applied mechanical stress versus the vertical displacement of the top rods (uniaxial stress) or the top ring (biaxial stress) for Si MOSFET samples ........... 26 2 3 Strain gauge calibration of the applied uniaxial mechanical stress versus the graduation of stress for GaN HEMT samples. ................................ .................... 27 2 4 Schematic illustration of conduction band structure change under <110> uniaxial tensile stress for bulk n type Si and Si (001) nMOSFETs. .................... 28 2 5 Schematic illustration of valence band structure change under <110> uniaxial compressive stress for bulk p type Si and Si (0 01) pMOSFETs. ........................ 30 2 6 The (001) 2D top band energy contours ( 25, 50, 75, and 100 meV ) with and without mechanical stress ................................ ................................ ................. 35 2 7 The (110) 2D top band energy contours (25, 50, 75, and 100 meV) with and without mechanical stress. ................................ ................................ ................. 36 2 8 Si and GaN subband E k diagrams. ................................ ................................ ... 37 2 9 A schematic of the conventional GaN HEMT structure. ................................ ...... 37 3 1 Strain induced channel resistance change of the (001)/<100> nMOSFETs under longitudinal, transverse, and biaxial tensile stresses. ............................... 43 3 2 Extracted coefficients of this work compared to literature for the (001) nMOSFETs ................................ ................................ ................................ ........ 44 3 3 Electric field dependence of biaxial coefficient for the (001)/<100> nMOSFETs. ................................ ................................ ................................ ........ 45 3 4 Extracted coefficients of this work compared to literature for the (001) pMOSFETs ................................ ................................ ................................ ........ 47 3 5 Illustrations of the TG FinFET structure. ................................ ............................. 49 3 6 The dependence of n channel TG FinFET coefficient s on the f in w idth. ........ 50 3 7 The dependence of p channel TG FinFET coefficient s on the f in w idth. ......... 51 PAGE 10 10 3 8 Experimentally observed d ependence of the coefficient s on gate overdrive for n channel TG FinFET. ................................ ................................ ................... 52 3 9 Experimentally observed d ependence of the coefficient s on gate overdrive for p channel TG FinFET. ................................ ................................ ................... 53 4 1 Illustrations of polarizations in the AlGaN and GaN layers. ................................ 59 4 2 Wurtzite GaN structures in the real and reciprocal spaces. ................................ 61 4 3 Lattice projection on the c plane of GaN under externally applied mechanical stress. ................................ ................................ ................................ ................. 63 4 4 Percent change of 2DEG sheet carrier density under uniaxial tension. ... 65 4 5 C hange of electron effective mass under external ly applied mechanical stress ................................ ................................ ................................ ................. 68 4 6 Change of GaN HEMT resistance (R TOT ) under longitudinal stress ................... 70 5 1 Schematic illustrations of the gate leakage process in a reverse biased GaN HEMT. ................................ ................................ ................................ ................ 75 5 2 Relation between the gate bias and the vertical electric field in the 18nm thick Al 0.26 Ga 0.74 N layer. ................................ ................................ ............................ 75 5 3 Simulation results of the FN tunneling current and the TFE current at room temperature. ................................ ................................ ................................ ....... 77 5 4 Schematic illustration of the 2 step bulk trap assisted leakage process. ............ 78 5 5 Simulation result of the 2 step bulk trap assisted leakage current at room temperature. ................................ ................................ ................................ ....... 80 5 6 Sche matic illustration of the Poole Frenkel Emission process from the surface states. ................................ ................................ ................................ .... 82 5 7 Modeling results of the Poole Frenkel Emis sion from surface states at various temperatures. ................................ ................................ ......................... 83 5 8 Change of the electron out of plane effective mass under uniaxial and biaxial stress. ................................ ................................ ................................ ................. 86 5 9 Change of polarization in the AlGaN layer under uniaxial and biaxial stress. ..... 87 5 10 Experimentally measured gate leakage current change under uniaxial tensile and compressive bending stress. ................................ ................................ ....... 88 PAGE 11 11 5 11 Schematic illustrations of the defect bond angle and bond length variation, and the defect level shift under lattice mismatch and wafer bending stress. ...... 90 5 12 Schematic illustration of the effect of externally applied mechanical stress on the r parameter. ................................ ................................ ................................ .. 90 5 13 Simulation results for the stress altered GaN HEMT gate leakage current at various temperatures. ................................ ................................ ......................... 91 6 1 The self consistent procedure for a standard DFT calculation. .......................... 94 6 2 Preparation of VASP input files. The basic input files are the POSCAR, POTCAR, INCAR, and KPOINTS file. ................................ ................................ 97 6 3 The calculated E k diagrams for bulk GaN. ................................ ...................... 101 6 4 The E k diagrams for bulk GaN. ................................ ................................ ....... 103 6 5 Literature published defect levels observed in GaN HEMTs. ........................... 105 B 1 The POSCAR file describing a 4 atom GaN unit cell. ................................ ....... 112 B 2 The POSCAR file describing a 32 atom GaN supercell. ................................ ... 113 C 1 The KPOINTS files used in the SC calculation step ................................ ........ 114 C 2 The KPOINTS file used in the non SC band structure calculation step. ........... 114 PAGE 12 12 LIST OF ABBREVIATION S CESL Contact etch stop layers DFT Density functional theory DOS Density of states FNT Fowler Nordheim tunneling GGA Generalized gradient approximation HEMT High electron mobility transistor HSE Heyd, Scuseria, and Ernzerhof LDA Local density approximation MOSFET Metal oxide semiconductor field effect transistor PAW Projector A ugmented W ave PBE Perdew, Burke, and Ernzerhof PFE Poole Frenkel Emission SCE Short chann el effect SSOI Strained silicon on insulator TFE Thermionic field emission TG FinFET Tri gate find shaped field effect transistor VLSI Very large scale integrated 2DEG Two dimensional electron gas PAGE 13 13 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CHARACTERIZATION AND MODELING OF STRAINED SI FET AND GAN HEMT DEVICES By Min Chu December 201 1 Chair: Scott E. Thompson Cochair: Toshikazu Nishida Major: Electrical and Computer Engineering Metal oxide semiconductor field effect transistors (MOSFETs) have shown impressive performance improvements over the past 10 years by incorporating strained silicon (Si) technology. Concurrently, interest in alternate device structure s and channel materials has been increas ing tremendously because of the scaling limitations on performance enhancement This work focuses on the impact of strain on state of the art Si planar MOSFETs, Si tri gate (TG) FinFETs, and GaN HEMT devices. Piezoresistive properties of Si n and pMOSFETs are obtained by applying controlled external mechanical stress, using the four point and concentric ring wafer bending setups The results are discussed by con sidering strain induced band splitting, band warping, and consequently the carrier repopulation, and the altered conductivity effective mas s and scattering rate. Strain experimental results on TG F in FETs coupled with the understanding of strained planar MOSFET physics are used to explain the strain enhanced tri gate device performance. Gallium Nitride (GaN) high electron mobility transistors (HEMTs) are promising for high power applications such as microwave or RF amplifiers. However the application PAGE 14 14 of these devices is limited by their reliability issues. A comprehensive study of the effects of mechanical stress on GaN HEMT channel resistance and gate leakage mechanisms is reported in this work. Using the tight binding method to calcul ate strained GaN band structure the stress altered channel resistance is simulated by considering two dimensional electron gas (2DEG) sheet carrier density and electron mobility variation. Several possible gate leakage mechanisms are modeled and co mpared to the experime ntal results The Poole Frenkel Emission from surface states is determined to be the dominant leakage mechanism and its stress dependence is investigated Finally, the density functional theory (DFT) calculations based on various functionals are evaluate d, and the HSE functional is employed to obtain the GaN band structure with correct bandgap PAGE 15 15 CHAPTER 1 STRAIN ENGINEERING IN STATE OF THE ART SI FET AND GAN HEMT 1.1 Motivation In the past three decades, scaling of MOSFETs has resulted in new technology generations every two to three years with doubled logic device density, lowered cost per logic function, and increased chip performance [1 3] However, as device dimension enters into the deep sub micrometer regime, many physical phenomena such as short channel effect (SCE), velocity saturation, high leakage current, and dielectric breakdown limit the benefit s of conventional scaling [4 8] To continuously improve device performance, new device structures, new materials, and strain engineering have been pr oposed and investigated. Among all these new technologies, strain engineering during the past decade has been the domin ant technique to enhance device performance while providing a low cost and low risk technique by maintaining the traditional MOSFET fabrication process. With the fourth generation [9] of strained Si technology now in commercial production, strai n enhanced performance and power saving are present in nearly all VLSI logic chips manufactured today As semiconductor device technology improves, strain is expected to be incorporated into new device structure s and channel material s to provide potential improvement To better predict the effect of strain in advanced technology, the strain enhanced Si planar MOSFETs need to be characterized first which serves as the foundation of understanding the role of strain in future technology. [2009] Annual Reviews. Chapter 1 is reprinted with permission from [M. Chu Y. Sun, U.Aghoram, S.E.Thompson Strain: A Solution for Higher Carrier Mobility in Nanoscale MOSFETs Annu. Rev. Mater. Res., vol. 39, April 2009 PAGE 16 16 Mu lti gate devices, such as TG FinFETs, have the advantages of better electrostatic, large ON/OFF current ratio, and lower power. However, the multiple gates have different surface orientations that respond differently to stress. The overall performance en hancement is not straight forward and needs to be studied. III V materials have the potential to replace Si in the future, due to its higher electron mobility as well as wider choice of bandgap Despite the GaN great potential for high power applic ation, its performance is limited by reliability issues such as gate leakage current hot electron injection, and barrier breakdown Understand ing the effect of stress o n the failure mechanism s is crucial since additional strain occurs in HEMT s during device operation, due to the inverse piezoelectric effect 1.2 Brief History of Strained Semiconductors 1.2.1 Strained Si FET Strain has been a topic of interest in semiconductor research since the 1950s. Bardeen & Shockley [10] developed deformation potential theory, which models the coupling between acoustic waves and electrons in solids, to calculate the components of the relaxation time tensor in terms of the effective m ass, elastic constants, and a set of deformation potential constants. In the deformation potential theory, the strain induced band edge shift is pr oportional to the strain tensor, (1 1) where are the deformation potentials. Herring & Vogt [11] generalized this theory in 1956 to model carrier transport in strained multi valley semiconductors and summarized a set of independent deformation potentials, and to chara cterize the conduction band valleys. Their work ascribed the electron mobility change to electron PAGE 17 17 transfer and an altered intervalley scattering rate caused by the valley energy shift. The picture of strain enhanced hole mobility is more complicated, ow ing to strong valence band warping. Thus, the hole transport under strain cannot be simply explained by band edge shift. Band structure calculation s such as the method [12 13] give more accurate valence band structure by constructing a strain Hamiltonian [14] in terms of the angular momentum derived by symmetry consideration. Hasegawa [15] and Hensel & Feher [16] used this method to systematically study the valence band effective masses and deformation potentials in strained Si. They revealed the key factor s that affect the hole mobility in semiconductors band splitting and warping, mass change, and consequently the change of DOS, which alters band occupation and phonon scattering. To date, deformation potential theory is still the primary method of modelin g the strai ned semiconductor and has prove n to be successful in explaining experimentally observed changes in device behavior under mechanical stress. The most effective empirical method to predict device behavior under strain is by measuri ng the piezores istance coefficients ( coefficients) [17] The first experimental data on coefficients for n and p type bulk Si and Ge were obtained by Smith in 1954 [18] These data have been used by the industry to model and predict MOSFET current enhancement under stress. However, it is inappropriate to use them to analyze MOSFET behavior in some cases because Smith s sample is bulk material without any surface confinement effect. In 1968, Colman et al. [19] measured the coefficients in p type inversion layers for the first time. Two decades later, the first Si n and pMOSFET s with biaxial stress induced by Si 1 x Ge x buffer layer ( Figure 1 1 A ) were demonstrated by Wesler et al. [20] in 1992 and by Nayak et al. [21 22] in 1994 PAGE 18 18 respectively. A 2.2 times enhancement in electron mobility and a 1.5 times enhance ment in hole mobility were reported. Rim et al. [23] investigated pMOSFET s drive current enhancement versus G e content in Si 1 x Ge x layers in 1995 and measured the current enhancement for short channel nMOSFETs in 1998 [24] In 2005, Lee et al. [25] published a review of the history of and progress in high mobility biaxially strained Si, SiGe, and Ge channel MOSFETs. A B Figure 1 1 Illustration of process induced stress on Si MOSFETs A) Biaxial stress caused from lattice mismatch be tween the Si channel and the relaxed SiGe substrate. B) Uniaxial stress induced by the nitride capping layer. PAGE 19 19 Even though the predominant focus of the industry in the 1980s and 1990s was on biaxially stressed devices, the current focus has shifted to uniaxial stress. Uniaxial stress has several advantages over biaxial stress, such as larger mobility enhancements and smaller shift in threshold voltage [26] Incorporating uniaxial stress to enhance MOSFET performance was first introduced by Ito et al. [27] and Shimizu et al. [28] who used etch stop nitride ( Figure 1 1 B ), and by Gannavaram et al. [29] who used SiGe source/drain (S/D) regions, in the early 2000s. Starting at the 9 0 nm technology node [30] uniaxial stress was successfully integrated into the mainstream MOSFET process flow to improve device performance [31 34] Encouraged by the strain enhanced planar MOSFETs, researchers recently applied uniaxial stress to multi gate devices [35 40] with metal gate and high k dielectric [41 42] as a performance booster. These studies demonstrate that strain a chieves higher Si FETs performance with extensive industrial application. 1.2.2 Strained GaN HEMT Unlike the strain technology in Si devices, stressing methods haven t been intentionally used during device fabrication to alter GaN HEMT performance. Nevertheless, the AlGaN layer is under large biaxial tensile stress due to its lattice mismatch from the bottom relaxed GaN substrate. In addition, since both the AlGaN and GaN are piezoelectric materials, the lattice deforms when an electric field is pre sent. Under typical GaN HEMT operation conditions, the total stress in the AlGaN layer is approximately 3GPa or higher This large amount of stress may vary the HEMT performance and reliability, and therefore has attract ed researchers attention in the p ast one decade. Gaska et al. [43] applied biaxial compressive stress to study its effect on channel electron densit y Their work observed decreasing electron density with PAGE 20 20 compressive stress. Eichhoff et al. [44] Kang et al. [45 46] Zimmermann et al. [47] Yilmazoglu et al. [48] and Koehler et al. [49] applied mechan ical stress to investigate the piezoresistive property of GaN HEMT s Large range of the extracted gauge factors was observed. Koehler et al. attributed this discrepency to the charge trapping effects during electrical measurement To investigate the effect of stress on HEMT reliability, del Alamo et al. [50 52] applied mechnical st ress while electrically stressing the devices. They concluded that external tensile stress increases the gate leakage cu rrent and decreases the critical breakdown voltage. Using the mechanical bending stress as a tool, more thorough studies of the strained GaN HEMT performance and reliability are still going on to achieve better understanding In the future, the viability of novel structures and channel materials will depend on their ability to provide device enhancement comparable to strained Si planar MOSFET s Thus, strain will remain a necessary enhancement option even in these devices. 1.3 Objectives and Organization In this work, the effect s of strain on channel resistance and reliability of semiconductor devices with different structures and channel materials are investigated. Chapter 2 discusses the fundamental physics regarding the str ained Si and GaN devices. In Chapter 3 a comprehensive set of two dimensional ( 2D ) inversion la yer coefficients of Si planar MOSFETs is extracted and compared to the published surface coefficients as well as the corresponding bulk Si values. A qualitative argument for the reported differences due to quantum confinement is given. B as ed on the knowledge of planar MOSFETs the measured l ongitudinal coefficient s of long channel TG FinFETs are then discu ssed Chapter 4 studies the strain altered channel resistance of PAGE 21 21 AlGaN/GaN HEMT, in terms of 2DEG sheet carrier density and electron mobility. Strain is incorporated into a tight binding model to simulate the gauge factor of HEMT devices, which will be compared to the experimentally extracted value to determine the best fit set of the elastic stiffness constants and piezoelectric coefficients of AlN and GaN. Chapter 5 addresses seve ral possible gate leakage mechanisms in the GaN HEMT The dominant mechanism is determ ined and its stress dependence is specified The DFT strategy for GaN band structure and defect calculation is discussed in Chapter 6. Finally, a summary of this research is p resented in C hapter 7 PAGE 22 22 CHAPT ER 2 FUNDAMENTAL PHYSICS BEHIND STRAINED SI AND GA N DEVICES Strain has been the main performance booster for Si MOSFETs in the past decade and has been intentionally incorporated into the device during fabrication. For the GaN HEMT, strain exists as an intrinsic device property due to lattice mismatch or arises f rom a non zero gate electric fi e l d due to the inverse piezoelectric effect. To better model and explain the behavior of strained Si FET and GaN HEMT devices, the fu ndamental strain physics behind these devices operating under conventional condition s needs to be clarified. This chapter targets a n overview of the strain and vertical electric field confinem ent effect s on Si and GaN device performance, from the energy b and structure perspective 2.1 Strain and Stress To obtain insights into the underlying physics of the strain enhanced device, it is necessary to first understand strain, of which the effect on current drivability is usually quantified as a coefficient. 2.1. 1 Strain Definition Strain can result from lattice mismatched film growth, phonon induced lattice vibrations, and applied external mechanical stress. Beneficial strain reduce s crystal symmetry, thus lifting band degeneracy and causing band warping. Any strain can be decomposed into a hydrostatic strain and two types of shear strain [53] For cubic crystals such as Si and Ge, hydrostatic strain does not break crystal s ymmetry and, hence, only shifts energy [2009] Annual Reviews. C hapter 2 is reprinted with permission from [M. Chu Y. Sun, U.Aghoram, S.E.Thompson Strain: A Solution for Higher Carrier Mobility in Nanoscale MOSFETs Annu. Rev. Mater. Res., vol. 39, April 2009] PAGE 23 23 levels without lifting band degeneracy. Thus, it is not important for carrier mobility enhancement. Large hydrostatic strain is undesirable owing to band gap narrowing, strain relaxation, and MOSFET threshold voltag e shifts. It is the shear component of strain that causes subband splitting and affects semiconductor transport propert ies Strain is introduced into the device channel preferably by applying uniaxial stress. The uniaxial stress is longitudinal when parallel to the channel and transverse when perpendicular to the channel. Large magnitudes of uniaxial channel stress (~ 1 GPa) are being incorporated in p channel devices of the 65 nm technology node [31 32] and an even higher stress level is appli ed in the 32 nm technology node, as is evident from the significantly large saturated drive current (1.55 mA/ m for NMOS and 1.21 mA/ m for pMOSFETs [9] ). However, since many process related parameters vary when fabricating strained MOSFETs, there is some uncertainty as to whether strain alone is responsible for the performance enhancement. To gain confidence in the e ffect of strain, external mechanical stress is applied using the four point or concentric ring wafer bending setup ( Figure 2 1 ). In this work, the drive current enhancement under both uniaxial and biaxial stress is studied using these setups to predict the device performance of strained Si planar MOSFETs, TG FinFETs, and GaN HEMT devices. 2.1 .2 Piezoresistance Coefficients The co efficient gives straightforward experimental information about strain enhanced carrier mobility in semiconductor s This coefficient is defined as the normalized change in resistivity per unit stress (2 1 ) PAGE 24 24 where is the external stress and is the resistivity, which can be calculated by (2 2 ) For MOSFETs under steady state, the electron and hole densi ties are approximately constant thus the coefficient is determined by the change in carrier mobility with stress. The coefficient gives us a straightforward idea about how much drive current enhancement can be achieved under particular stress, and therefore has been widely used in industry to predict strained device performance [54 56] For cubic symmetric material such as Si, the longitudinal and transverse coefficients ( l and t ) of any direction can be calculated from the piezoresistance tensor and direction cosines [57] and the bi axial coefficient ( B ) equals l + t When a vertical electric field is applied (MOSFET cases ), l t and B may vary from their bulk value due to quantum confinement. 2.1.3 Wafer Bending The standard four point bending setup [58] is used to generate a uniaxial stress at the top surface of a rectangular waf er piece, as shown in Figure 2 1 A For Si samples, t he surface stress along the uniaxial stress direction ( perpendicular to the rods ) is calculated from the relation [59] (2 3 ) where E stress direction, Z is the sample vertical displacement, t is the thickness of the sample, a is the distance between the inner and outer rods, and L is the distance between the two outer rods. This calculated uniaxial PAGE 25 25 stress showed good agreement with the independent measurement of a strain gauge mounted on the Si wafer surface ( within 5% error to the actual stress measured by the resist ance change in the st rain gauge) as shown in Figure 2 2 A B Figure 2 1. Illustrations of mechanically bended wafe r samples. A) Four point wafer bending for applying uniaxial stress. B) C oncentric ring wafer bending for applying biaxial stress [2008] American Institute of Physics [reprinted with permission from M. Chu et al., Comparison bet ween high field piezoresistance coefficients of Si metal oxide semiconductor field effect transistors and bulk Si under uniaxial and biaxial stress J. Appl. Phys., vol. 103, pp. 113704, Fig. 3 and Fig. 5, June 2008] The rods are replaced by two concentric rings of diffe rent radii to generate a biaxial stress on a Si wafer piece, as shown in Figure 2 1 B Finite element method was PAGE 26 26 used to simulate the applied biaxial stress using ABAQUS and two experimental methods ( optical strain characterization from the measured wafer curvature and direct measurement using a strain gauge ) were used to measure the induced biaxial stress versus the rings vertical displacement The resulting stress calibration of the biaxial bending setup is shown in Figure 2 2 G ood agreement is ac hieved between the Finite Element A nalysis and the two independent measurements and therefore this stress configuration will be used to calculate the biaxial coefficients. Figure 2 2 Applied mechanical stress v ersus the vertical displacement of the top rods ( uniaxial stress ) or the top ring ( biaxial stress ) for Si MOSFET samples Stress calibration is bas ed on three different methods [2008] American Institute of Physics reprint with permission from [M. Chu et al., Comparison bet ween high field piezoresistance coefficients of Si metal oxide semiconductor field effect transistors and bulk Si under uniaxial and biaxial stress J. Appl. Phys., vol.103, pp.113704, Fig. 4, June 2008] PAGE 27 27 For the GaN HEMT wafer samples, the strain gauge measurement is carried out to calibrate the 4 point wafer bending induced uniaxial stress and the result is shown in Figure 2 3. Figure 2 3 Strain gauge calibration of the applied uniaxial mechanical stress versus the graduation of stress for GaN HEMT samples. 2.2 Strained Si devices Strain has a larger effect on the current conduction in Si than on a metal, owing to the fact that not only do the physical dimensions change under strain but the carrier mobility can also be enhanced. The e ffects of s train on ca rrier mobility in Si devices have been intensively studied over the past 10 years. Previous works [11, 60 62] pointed out that the main factors affect ing mobility are ( 1 ) the change in the average effective mass due to carrier redi stribution or band warping and ( 2 ) the change in the carrier scattering rate due to energy band splitting or density of state variation. Since PAGE 28 28 the applied vertical electric field in MOSFETs creates a potential well that confines the carrier in the transve rse direction, the allowed energy levels are quantized. Therefore, in inversion mode devices, the net energy band splitting is due to the cumulative effect of both strain induced band splitting and 2D electrostatic surface confinement induced band splitti ng [60] Furthermore, since confinement is a function of the transverse electric field caused by the applied gate voltage, in general, the surface coefficients may not be the same as the bulk Si coefficients. 2.2 .1 Stress Effects on N Si Band Structure Figure 2 4. Schematic illustration of conduction band structure change under <110> uniaxial tensile stress for bulk n type Si and Si (001) nMO SFETs. PAGE 29 29 The Si condu ction band minima are located near the X point. Due to its crystal symmetry along the [100] major axis, there are six equivalent constant energy ellipsoids around each minimum. When external stress is present, the strain induced conduction band edge shift is given by, (2 4 ) w here and are the dilation and shear deformation potential s of the conduction band, respectively, i refers to one of the six valleys, is the trace of the strain tensor and is a unit vector in the reciprocal space [53] The band edge shift can also be expressed in terms of hydrostatic band edge shift and shear band edge splitting by (2 5 ) where only the splitting term causes carrier redistribution and thus mobility change. For example, the <110> longitudinal tensile stress causes the energy of the 2 subband to shift down and the energy of the 4 to shift up (the separation between 2 and 4 is calculated to be 40 meV under 1GPa using Eq. ( 2 4 ) ), resulting in electrons repopulating from the 4 valley to the 2 valley, as shown in Figure 2 4 Because th e conductivity effective mass of the 2 valley (0.19m 0 ) is smaller than that of the 4 valley (0.315m 0 calculated using the method in [63] ), the electron repopulation into the 2 valley causes the average effective mass to decrease and carrier mobility to increase. The band splitting also alters the scattering rate The dominant scattering mechanisms in strained n Si devices are intervalley phonon scattering [64] and surface roughness scattering [65 ] As the six fold conduction band s split, the intervalley scattering rate PAGE 30 30 becomes lower owing to the smaller DOS [60] thus resulting in a higher mobility. A more complete discussion of phonon limited electron mobility enhancement is provided by Takagi et al. [61] Figure 2 5 Schematic illustration of valence band structure change under <110> uniaxial compressive str ess for bulk p type Si and Si ( 001) pMOSFETs. Strained nMOSFETs have different mobility enhancement factors from those of bulk Si. Because of the electric field confinement, the 2 and 4 valleys are originally non degenerate for unstrained Si. The energy splitting between these valleys depends on the magnitude of the electric field and the difference in their out of plane confinement effective mass It can be calculated from a self consistent solution of the Schrdinger and Poisson equations [66] Under low electric field, the band splitting without stress is PAGE 31 31 small, and thus the electron distrib ution does not differ much from the bulk Si. In contrast, as s hown schematically in Figure 2 4 the band splitting without stress is already large under high electric field (>0.7 MV/cm), causing over 80% of electrons to be locate d in the low energy valleys. Thus, further change in the average effective mass under stress is very small (< 10% ) [60] Furthermore, surface roughness scattering dominates under typical commercial use co nditions for two dimensional (2 D) carrier transport, which makes the current transport in nMOSFETs more difficult to predictably model [67 68] As a result, the high field coefficients can differ significantly from the bulk coefficients but must converge to the bulk value as the transverse field goes to zero. 2.2 .2 Stress Effects on P Si Band Structure The heavy hole (HH) and light hole ( LH ) valence band minima are degenerate at the point for bulk Si In unstressed Si, 80% of the holes occupy the HH band, which has an effective mass of 0.59 m 0 along the < 110 > direction (versus 0.15 m 0 for the LH band). Compared to the n Si band structure, the valence bands warp significantly under stress, which modifies the in plane effective mass and the DOS [69] Figure 2 5 schematically illustrates the valance band warping under <110> uniaxial compression. In the < 110 > direction, the HH band ( top band ) becomes LH like around the point, and the LH band ( bottom band ) becomes HH like. The top band is lower in energy and has smaller co nductivity effective mass (0.11 m 0 for 1 GPa), whereas the bottom band is higher in energy and has larger conductivity effective mass (0.2 m 0 for 1 GPa). Most of the holes therefore repopulate into the top band, and the mobility is enhanced. As a result, not only the band edge but also the band structure close to the point is PAGE 32 32 important for hole transport owing to the significant band warping In addition, the HH, LH, and spin orbit split off hole bands are energetically close to each other, which makes the valence band structure calculation more complex. The valence band structure of bulk p Si under stre ss can be numerically estimated, in particular, near the band edge using the k p method with the Luttinger Kohn strain Hamiltonian [13, 70] A t room temperature, band warping induced effective mass reductio n is the dominant factor for mobility enhancement in p type Si under uniaxial stress ( < 1 GPa) [62] T he stress induced phonon scattering rate change is negligible [69] since the splitting between the top band and the bottom band is small compared with the Si optical phonon energy (61.3 meV). For unstrained pMOSFETs, however, the degeneracy of the heavy hole and the light hole is lifted by the surface electric field con finement, as shown in Figure 2 5 Similar to the conduction band, the valence band structure can be calculated by solving the Schrdinger and Poisson equations self consistently. Compared with the bulk Si energy contours, the transverse electric field splits the bands but does not modify the in plane subband structure for the (001) surface devices, as shown in Figure 2 6 [69] Therefore, the strain induced hole effectiv e mass change is expected to be similar for bulk p Si and Si pMOSFETs. However, the in plane DOS changes with stress (i.e., becomes smaller under uniaxial compression), which can cause a phonon scattering rate variation. We neglect this effect in bulk p Si because the strain induced band splitting (~ 20 meV for 1 GPa) is small compared to the optical phonon energy for Si For the (001) oriented p MOSFETs under typical operation condition (surface effective field E eff ~ 1MV / cm), the summation of confinement induced band splitting and strain induced PAGE 33 33 band splitting can be larger than the Si optical phonon energy and the phonon scattering change cannot be neglected. As a result, the (001) p MOSFET s coefficients depend on both the DOS alteration a nd effective mass vari ation. For the (110) oriented p MOSFETs, the electric field significantly modifies the in plane band structure, as shown in Figure 2 7 which results in totally different mobility enhancement, and thus a variation of the 2D coeffici ents from the corresponding bulk values [69] 2.3 Strained GaN devices Following the same process as strained Si analysis, the band structure of wurtzite GaN is the start ing point of investigating strain related GaN HEMT behavior. Nido [71] and Jogai [72] incorporated external stress into an empirical sp 3 d 5 tight binding model [73] to study the consequen ce of strain effects on the GaN band structure. Their work, how ever, considered only biaxial stress and focused mainly on the bandgap and valance band structure. Since the electron is the majority carrier in the GaN HEMT channel, it is essential to study the strain altered conduction band. In addition, uniaxial stres s, which is the most beneficial stress for Si devices, is worth considering in GaN. Electron mobility altered by uniaxial stress is the main focus of this part of my research. Similar to Si, e lectron mobility enhancement for bulk GaN can result from the a verage conductivity effective mass redu ction and a suppression of int ra valley scattering. Stress affects electron effective mass through two factors: band splitting induced electron repopulation and band warping. Unl ike Si which has six degenerate conduction bands, or GaAs which has energetically adjacent conduction bands, GaN is a direct band gap material with only one conduction band as shown in Figure 2 8 As a result, no electron repopulation occurs under stress, and thus band warping is the only PAGE 34 34 mechanism responsible for effective mass change. The absence of conduction band splitting also results in negligible change of the acoustic phonon scattering The polar optical phonon scattering also has negligible dependence on stress, due to the f act that the mechanical stress does not induce polarization change along the longitudinal direction Therefore, the change in effective mass through band warping is the dominant mechanism for stress dependent bulk GaN electron mobility variation. In a con ventional GaN HEMT structure a thin layer ( 10~30 nm) of AlGaN is deposited on top of a relatively thick layer ( 2~5 m) of GaN as shown in Figure 2 9 Due to the lattice mismatch between the AlGaN and GaN, t he AlGaN layer is then considered biaxially stressed while the GaN layer is relaxed. Since both the AlGaN and GaN are piezoelectric materials, polarization appears in both layers. The total polarization difference between them gives rise to the 2DEG that forms on the GaN side of the interface between the AlGaN and GaN layers For an Al fraction of 0.26, the 2DEG density is approximately 1 10 13 /cm 2 As a result, the channel forms even at zero gate bias and the GaN HEMT has negative threshold voltage. T he 2DEG is confined at the interface and electron energy is quantized. However, t here will not be confinement induced sub band splitting in GaN HEMT since GaN has only one single conduction band. Therefore similar to bulk GaN, the scattering rate change in strained GaN HEMT re mains negligible and the band warping induced effective mass variation should dominate its stress behavior The detail ed modeling of strain altered electron effective mass through conduction band warping will be discussed in Chapter 4 PAGE 35 35 A B Figure 2 6 The (001) 2D top band energ y contours ( 25, 50, 75, and 100 meV ) with and without mechanical stress A) B ulk p type Si B) T he (001) p MOSFETs [ Reprint ed with permission f rom Sun G. 2007. PhD dissertation (Page.68 69, Figure 3 20 and 3 2 1). University of Florida, Gainesville, Florida ] PAGE 36 36 A B Figure 2 7 The (110 ) 2D top band energy contours (25, 50, 75, and 100 meV) with and without mechanical stress. A) B ulk p type Si B) T he ( 110 ) p MOSFETs. [Reprint ed with permission f rom Sun G. 2007. PhD dissertation (Page.70 71, Figure 3 22 and 3 23). University of Florida, Gainesville, Florida ]. PAGE 37 37 A B Figure 2 8 Si and GaN s ubband E k diagrams A) For bulk S i [reprinted with permission from [74] Page 81]. B) For bulk GaN. Si has six degenerate conduction bands with minima locate near the X point. GaN has single condu ction band locates at the point Figure 2 9 A schematic of the conventional GaN HEMT structure. The band profile along the vertical direction (gate AlGaN GaN) is shown on the right side. PAGE 38 38 CHAPTER 3 PIEZORESISTANCE OF S I DEVICES 3.1 Introduction Strained silicon is a preferred feature enhancement for high performance Si logic technology due to its advantage of enhancing channel carrier mobility and hence improving MOSFET performance [31, 3 4, 54, 75 76] Since the 1990s, the physical understanding and modeling of carrier mobility enhancement due to strain has been aggressively studied for both n type [60 61, 77] and p type [60, 62, 65, 78 79] Si planar MOSFET inversion layers. The coefficient, which is defined as the normalized change in resistivity with applied stress, is one of the most useful parameters to capture and model the strain altered current. The original coefficients for silicon and germanium were determined for bulk silicon and germanium by Smith [18] 50 years ago. Although the bulk Si coefficient values provide useful qualitative insight into the behavior of strained MOSFETs [54 56] it has been shown [19, 80 85] that 2D surface confinement effects caused by the vertical electric field, such as subband splitting and band warping, alter some of the coefficients. These works considered only uniaxial coefficients which were extracted under relatively low electric field (< 0.7 MV cm). It is only recently that the high field coefficients (obtained from carrier mobility enhancement), under both uniaxial and biaxial stresses, were studied [86] However, there is a need for a comprehensive comparison of confined coefficients under high [2008] American Institute of Physics Chapter 3 is reprinted with permission from [M. Chu T. Nishida X.L. Lv N. Mohta S.E.Thompson Comparison bet ween high field piezoresistance coefficients of Si metal oxide semiconductor field effect transistors and bulk Si under uniaxial and biaxial stre ss J. Appl. Phys., vol.103, pp.113704, June 2008] PAGE 39 39 electric field for both n and pM OSFETs under uniaxial and biaxial stresses for the (001) and (110) substrates which is one of the focuses of this chapter Three dimensional multi gate (MG) transistors [87] such as tri gate (TG) Fin FETs [88] are potential alternatives to replace conventional MOSFETs at the 22nm node and beyond, because of their superior control of the Short Channel Effects (SCE) [87] Meanwhile, strain has been widely used to enhance bulk Si MOSFET drive current [54] These two technologies have been recently combined and additional performance enhancement in MG devices is achieved via process induced strain, such as Si x Ge 1 x SiC in so urce /drain regions [36, 89 90] strained nitride contact etch stop layers (CESL) [91] supercritical strained silicon on insulator (SC SSOI) [37, 92 93] and strained SiGe on insulator (SSGOI) [35, 94] However, since many process flow parameters are changed when incorporating strain there is some uncertainty if strain alone is responsible for the performance enhancement. External mechanical stress applied by wafer bending is a direct method to study strain effect and reveal the underlining physics. Biaxial wafer bending on TG FinFETs was reported in [38] but few works investigate the effect of uniaxial stress, which is the proper stress for simultaneously improving n and p channel TG Fin FET performance [40] In this chapter four point wafer bending setup is used to apply uniaxial stress in the channel of TG Fin FET The longitudinal coefficient s and their dependence on fin width and gate bias are extracted. A model is then proposed and validated to predict strained TG Fin FET current enhancement. PAGE 40 40 3.2 P iezoresistance of P lanar n and p MOSFETS 3.2 1 Device and Experiment Details In order to extract the 2D inversion layer coefficients, the strain altered channel resi sta nce was measured using n and p MOSFETs on two surface orientations, ( 001 ) and ( 110 ) with two channel directions, namely (001)/<100>, (001)/<110> (110)/<100>, and (110)/<110>. The details of the measured MO SFET devices are as follows: 10 m 1 0 m channel length/width, 1.2 1.4 nm gate oxide, n+/p+ polysilicon gates for n /p MOSFETs, and ~ 10 18 /cm 3 channel doping density. The channel resistances were measured by using a Keithley 4200 semiconductor parameter analyzer, under a gate effective field of E eff ~1 MV/ cm and a drain voltage of  V ds =50 mV. There are two major sources of uncertainty during coefficient extraction: the uncertainty of applied stress and the uncertainty of device measurement. The total uncertainty of the extracted coefficients can be estima ted by combining both factors. (3 1) (3 2) (3 3) w here is the unc ertainty of electrical measurement, is the uncertainty of the applied stress, is the sensitivity coeffic ient of device measurement, and is the sensitivity coefficient of the applied stress. To calculate both and must be investigated. In this study, is estimated as the 95% confidence interval of the R R measurements under stress. PAGE 41 41 is the uncertainty of each applied stress, which is estimated to be 1/5 of the minimum stress interval ( 4 MPa ) used in this work. Using this process, the uncertainty estimates are listed in Table 3 1 for nMOSFETs and Table 3 2 for pMOSFETs, along with the c orresponding extracted coefficients. 3.2.2 Results and Discussion The st ress altered channel resistance ( R/R ) of the (001) surface < 100 > long channel n MOSFETs is shown in Figure 3 1 including results under in plane longitudinal, transverse uniaxial tensile stress, and biaxial tensile stress. Neglecting the geometrical contribution to R R the coefficients are extracted by dividing R R by the applied uniaxial stress calculated using Eq. ( 2 3 ) or biaxial stress which is experimentally determined using the calibration methods from Fig ure 2 2 Following the same process, the coefficients for both n and p MOSFETs on the ( 001 ) / ( 110 ) oriented surface with the < 100 > /< 110 > channel directions are ext racted and summarized in Table 3 1 and Table 3 2 and compared with the bulk Si values from Smith. 3. 2.2. 1 Stress results in n type devices Table 3 1 summarizes the extracted coefficients for n MOSFETs. The result s show that the coef ficients of ( 110 ) oriented n MOSFETs are significantly different from the corresponding bulk value. For the ( 001 ) oriented devices, however, the 2D coefficients are close to the bulk value for some particular cases. The uniaxial coefficients show good agreement between the ( 001 ) / < 110 > n MOSFET s while there is a large difference in the case of the ( 001 ) / < 100 > device (i.e. l = 102/ 47, t = Si and n MOSFETs, respectively ) This difference is attributed to two factors: the high channel doping PAGE 42 42 density and the transverse electric field confinement. It is shown in [95] that the electron mobility anisotropy decreases at high doping concentrations due to the increased ionized impurity scattering. Therefore, the electron repopulation under stress does not affect the average mobility as much as in the lowly doped sample. In addition, most electrons are already located at the lowest energy valleys under electric field confinement, which results in even lower strain induced effective mass change. In this case, variation of carrier scattering rate domina tes the total mobility change. As a result, the uniaxial coefficients of the ( 001 ) / < 100 > n MOSFETs can be significantly smaller than the bulk value depending on its actual doping density and the applied gate voltage. Figure 3 2 compares the extracted un iaxial coefficients of this work to the literature. It confirms that the ( 001 ) / < 110 > n MOSFET s coefficients are relatively insensitive to doping density and quantum confinement based on the similar observations of many different devices. It also indic ates a much wider variation for the ( 001 ) / < 100 > device with higher doping density and effective field (~ 1 MV/cm ) The l and t can become smaller and, in fact, the t can change its sign under high effective field if the scattering rate change is greater than the effective mass change. For the ( 001 ) oriented n MOSFETs under biaxial stress, the coefficient is not much different from the bulk value, as shown in Table 3 1 This can be understood by the fact that the elect ric field confinement effect on l and t cancel each other [82] Since B = L + T the offset of confinement altered L and T result s in a B that is almost independent of the vertical electric field Figure 3 3 shows the measured ( 001 ) B versus electric field, indicating little field dependence. PAGE 43 43 Figure 3 1 Strain induced channel resistance change of the ( 001 ) / < 100 > n MOSFETs under longitudinal, transverse, and biaxial tensile stresses. The applied gate effective field is 1 MV/cm. [2008] American Institute of Physics reprint ed with permission from [ M. Chu et al. Comparison bet ween high field piezoresistance coefficients of Si metal oxide semiconductor field effect transistors and bulk Si under uniaxial and biaxial stress J. Appl. Phys., vol.103, pp.113704, Fig. 6, June 2008] Table 3 1. Experimental extracted coefficients of Si planar nMOSFETs They are compared to the corresponding bulk n Si value from Smith. ( Number in parentheses is the estimated uncertainty.) [2008] American Institute of Physics reprint ed with permission from [ M. Chu et al. Comparison bet ween high field piezoresistance coefficients of Si me tal oxide semiconductor field effect transistors and bulk Si under uniaxial and biaxial stress J. Appl. Phys., vol.103, pp .113704, Table I, June 2008] l t B (001) Surface <100>Channel n MOSFET s 47 (7.7) 22 (4 .0 ) 50 (2.3) Smith bulk Si 102 53 49 (001) Surface <110>Channel n MOSFET s 32 (7.4) 15 (6.4) 47 (3.2) Smith bulk Si 31 18 49 (110) Surface <100>Channel n MOSFET s 24 (1 .0 ) 25 (1 .0 ) 10 (2.4) Smith bulk Si 102 53 49 (110) Surface <110>Channel n MOSFET s 3 7 (1.8) 11 (2.4) 7 (3.9) Smith bulk Si 31 53 49 PAGE 44 44 As a conclusion the ( 001 ) oriented n MOSFETs under biaxial stress have the largest coefficients which have little dependence on the vertical electric field. Results in Table 3 1 indicate that the surface confined coefficients are not always the same as used to analyze highly doped ( 001 ) / < 100 > n MOSFETs operating under high vertical effective field. Figure 3 2 Extracted coefficients of this work compared to literature for the ( 001 ) nMOSFETs [18, 55, 80, 82, 84, 86, 96] T he applied gate effective field is 1 MV/cm in our measurement. [2008] American Institute of Physics reprint with permission from [ M. Chu et al. Comparison bet ween high field piezoresistance coefficients of Si metal oxide semiconductor field effect transistors and bulk Si under uniaxial and biaxial stress J. Appl. Phys., vol.103, pp.113704, Fig. 7, June 2008] PAGE 45 45 Figure 3 3 Electric field dependence of biaxial coefficient for the ( 001 ) / < 100 > nMOSFETs. [2008] American Institute of Physics reprint with permission from [ M. Chu et al. Comparison bet ween high field piezoresistance coefficients of Si metal oxide semiconductor field effect transistors and bulk Si under uniaxial and biaxial stress J. Appl. Phys., vol.103, pp.113704, Fig. 8, June 2008] 3. 2.2 .2 Stress results in p type devices Table 3 2 summarizes the extracted coefficients of Si p MOSFETs and the corresponding bulk value from Smith. The result s show that the coefficients of ( 110 ) oriented p MOSFETs are completely different from the corresponding bulk value. For the ( 001 ) oriented devices, however, the 2D coefficients agree with the bulk value for most cases. Table 3 2 also lists the theoretical calculation results from [62] which have considered various surface/channel orientations, effective mass effect, phonon scatt ering effect, and surface roughness scattering effect. The theoretical results show good overall agreement with our experiment data, and thus confirm that the physical PAGE 46 46 model of [62] can be used to explain the di fference between p MOSFET s and bulk p Si coefficients. Table 3 2. Experimental extracted coefficients of Si planar p MOSFETs They are compar ed to the corresponding bulk p Si value from Smith and the simulated value from [62] ( Number in parentheses is estimated uncertainty.) [2008] American Institute of Physics reprint with permission from [ M. Chu et al. Comparison bet ween high field piezoresistance coefficients of Si metal oxide semiconductor field effect transistors and bulk Si under uniaxial and biaxial stress J. Appl. Phys., vol.103, pp.113704, Table II, June 2008] l t B (001) Surface <100>Channel p MOSFET 15 (6.4) 9 (4.3) 11 (3.7) Smith bulk Si 6.6 1.1 5.5 (001) Surface <110>Channel p MOSFET 71 (15.6) 72.7 [62] 32 (7.7) 45.8 [62] 16 (5.8) Smith bulk Si 71.8 66.3 5.5 (110) Surface <100>Channel p MOSFET 31.3 (1 .0 ) 9.5 (1 .0 ) 25.4 (2.4) Smith bulk Si 6.6 1.1 5.5 (110) Surface <110>Channel p MOSFET 27 (8.8) 39 [62] 5 (3 .0 ) 6.6 [62] 25.8 (2.2) 28.7 [62] Smith bulk Si 71.8 1.1 5.5 Our result s show that the ( 001 ) / < 110 > p MOSFET s have the largest coefficients, followed by the ( 110 ) substrate devices, and then the ( 001 ) / < 100 > devices. The coefficients for the ( 001 ) oriented p MOSFETs, except the ( 001 ) / < 110 > transverse result, agree with the bulk value and have little vertical electric field depend ence [86] This can be understood from the complementary effect s of DOS alteration on reducing the scattering and reducing the effective mass for these stresses [62] For the ( 001 ) / < 110 > p MOSFETs under transverse stress, in contrast, these two effects are subtractive ( i.e., increased effective mass with decreased scattering under transverse compressive stress ) which results in a smaller mobility change than bulk p Si. As a result, the corresponding coefficient is smaller than the bulk value. Detailed discussion of the strain induced change in the hole mobility can be found in [62] In PAGE 47 47 general, while the strain induced band warping, band splitting, and 2D DOS alteration alter the effective mass and scattering rate, the confinement induced band s plitting also affects carrier distribution and scattering. Both of these effects must be included to investigate the strain effect on p MOSFETs. Figure 3 4 Extracted coefficients of this work compared to literature for the ( 001 ) pMOSFETs [18 19, 55, 80, 82, 84, 86, 96] The applied gate effective field is 1 MV/cm in our measurement. [2008] American Institute of Physics reprint with permission from [ M. Chu et al. Comparison bet ween high field piezoresistance coefficients of Si metal oxide semiconductor field effect transistors and bulk Si under uniaxial and biaxial stress J. Appl. Phys., vol.103, pp.113704, Fig. 9, June 2008] Figure 3 4 compares the extracted pMOSFET s uniaxial coefficients of this work to the literature which shows good overall agreement. For p MOSFETs under a PAGE 48 48 relatively low biaxial stress (< 200 MPa ) the coefficients are much smaller than the ( 001 ) / < 110 > uniaxial va lue, because of the smaller effective mass change under biaxial stress than uniaxial stress [69] Therefore, only small current enhancement is expected for biaxial stressed devices and uniaxial stress is more advantageous for p MOSFETs. In summary, our result in Table 3 2 shows that the ( 001 ) / < 110 > p MOSFETs under longitudinal stress have the largest coefficients with little vertical electric field dependence to analyze the ( 001 ) oriented p MOSFETs, except for the ( 001 ) / < 110 > transverse stress. For devices on the ( 110 ) substrate, surface confinement must be taken into account. 3.3 Piezoresistance of TG FinFETs 3.3.1 Device and Experiment Details The TG Fin FETs were fabricated on (001) oriented SOI wafers with 145nm buried oxide (BOX) and 65nm starting silicon layer [37] Fins with widths varying from 10um down to 20nm were patterned using 193nm lithography. The fins were left un doped. The gate stack consists of a 2nm thick HfO 2 dielectric deposited by atomic layer chemical vapor deposition (ALCVD) and a 5nm TiN layer that is capped with 100nm polysi licon. N iSi was used as a salicide. Fig ure 3 5 A shows the TEM image of the finished device. The fin length L width W and fin height H are defined as shown schematically in Fi g ure 3 5 B All devices have <110> channel direction and (110) side wall. Uniaxial stress was applie d via 4 point wafer bending [83] for 10um long TG Fin FETs with varying fin widths and the coefficients were extracted under a gate over drive ( V G V TH ) ranging fr om 0.2V~0.7V. PAGE 49 49 3.3.2 Experiment al Results In this section, experiment al results of the extracted coefficients for both n and p channel TG Fin FETs are reported. Fig ure 3 6 and Fig ure 3 7 show the coefficient dependence on the fin width of n and p channel TG Fin FETs respectively With decreasing fin width, it is observed that the coefficients increase slightly from 29 to 38 for n channel devices, whereas they decrease dramatically from 53 to 25 for p channel devices. Table 3 3 summarizes th e coefficient s for TG Fin FETs with W=10um and 0.02um, along with the ( 001 ) and (110) planar MOSFETs value s [17] as well as the results reported in [39] for DG FinFETs. A B C D Fig ure 3 5 Illustrations of the TG FinFET structure. A ) TEM image of a 20nm wide Fin B) S chematic structure of a Fin with W=H C) S chematic structure of a Fin with W>>H D) S chematic structure of a Fin with W< PAGE 50 50 Fig ure 3 6 The dependence of n channel TG FinFET coefficient s on the f in w idth. The solid curve is the theoretical value calculated by weighted averaging ( 001 ) and (110) surface coefficient s with respect to fin width and fin height. Table 3 3 The longitudinal coefficient s ( 10 12 dyne/cm 2 ) for n and p channel TG Fin FETs Our extracted results are compared to the literature published values for the ( 001 ), (110) planar MOSFETs and DG FinFETs. Device Type P channel N channel TG FinFET (W = 10 um) 53 29 TG FinFET (W = 0.02 um) 25 38 Planar MOSFETs [17] (001) surface: 71.8 (001) surface: 31.5 (110) surface: 27.3 (110) surface: 37.0 DG FinFETs [39] 37 51.4 The coefficient of n channel TG Fin FETs with W=10um ( 29) is close to the ( 001 ) planar nMOSFET s value ( 31.5 [17] ). F or n channel device with W=0.02um, the coefficient ( 38) is the same as the (110) planar nMOSFET s value ( 37 [17] ), but PAGE 51 51 smaller than the DG n FinFET value ( 51.4 [39] ). The coefficient of p channel TG Fin FETs with W=10um (53) however, is smaller than the ( 001 ) planar p MOSFET s value (71.8 [17] ). F or p channel TG device with W= 0.02u m the coefficie nt (25) is similar to the (110) planar p MOSFET s value (2 7 .3 [17] ) but smaller than the DG p FinFET value (37 [39] ) Fig ure 3 7 The dependence of p channel TG FinFET coefficient s on the f in w idth. The solid curve is the theoretical value calculated by weighted averaging ( 001 ) and (110) surface coefficient s with respect to fin width and fin height. The TG Fin FETs coefficient dependence on the gate overdrive voltage is also extracted, and the results for n and p channel devices are plotted in Fig ure 3 8 and Fig ure 3 9 respectively. Fig ure 3 8 shows that the gate bias has negligible effect on PAGE 52 52 the coefficient for both wide and narrow n channel TG Fin FET. The variation is less than 5%. For p channel TG Fin FET coefficient as shown in Fig ure 3 9 however, strong dependence on gate bias is observed in wide width devices, whereas no evidence of gate bias dependence is seen in narrow width device. Fig ure 3 8 Experimental ly observed d ependence of the coefficient s on gate overdrive for n channel TG Fin FET. 3.3.3 Discussion By investigating the coefficient dependence on the fin width and the gate overdrive voltage, the strain enhanced TG Fin FET behavior can be explained and predicted. The experiment al results are compared to the theoretical expectation, and the discrepancy is discussed in this s ection PAGE 53 53 Fig ure 3 9 Experimentally observed d ependence of the coefficient s on gate overdrive for p channel TG Fin FET. 3.3.3 .1 A m odel for s train e nhanced TG Fin FETs The experiment al results can be explained by a theoretical model which weighted avera ges the (00 1 ) and (110) surface coefficients as shown in Eq. 3 4. (3 4) From this model, when W >> H (Fig ure 3 5 C ), current conduction on the top surface dominates and the stressed TG Fin FET behavior is expected to be the same as a ( 001 ) surface strained planar MOSF ET. In contrast, when W << H (Fig ure 3 5 D ), it is expect ed that the strained TG Fin FET behavior is the same as a (110) surface strained planar PAGE 54 54 MOSFET because the (110) side wall conduction dominates. For very thin W ( W <20nm), bulk inve rsion become significant [97 98] and the TG Fin FET should show similar strain induced enhancement as a DG FinFET. 3.3.3 .2 N channel TG Fin FET b ehavior Over the entire fin width range down to 20nm, the experimental extracted coefficient s agree with the the oretical model, as shown in Fig ure 3 6 Furthermore, the TG Fin FET coefficient s have negligible dependence on gate overdrive, which is also similar to the coefficient s of strained planar n channel MOSFETs with <110> channel. This is because the 2 subband warping under <110> uniaxial stress does not depend on the co nfinement indu ced splitting [60] As a result, for n channel TG Fin FETs down to 20nm in width strained current enhancement can be modeled by the ( 001 ) and (110) planar MOSF ET coefficient s, with no observation of higher coefficient value caused by bulk inversion as in DG FinFET. 3.3.3 .3 P channel TG Fin FET b ehavior The extracted coefficient s match the theoretical model for narrow width TG Fin FETs where side wall conduction dominates. However, the data for wide width TG Fin FET is smaller than the theoretical value. This discrepancy can be understood by the gate overdrive dependence of the TG Fin FET coefficient as plotted in Fig ure 3 9 Th e figure shows a decreasing coefficient versus increasing gate overdrive for large width devices and a constant coefficient for devices with small fin width. Because of the un doped fin body and the small electric oxide thickness, inversion carrier de nsity increases rapidly under gate bias and can reach a level over 1.5 10 /cm The simulation done in [69] indicates that the coefficient for ( 001 ) planar MOSFET s PAGE 55 55 becomes less than 2/3 of its original value at this density level and beyond, which matches our result for large width TG Fin FET s For small fin width, however, the coefficient keeps relatively constant even with high carrier density. This is because the subband sp litting between the top two val e nce bands for (110) s urface is so large (>60 meV [62] ) that the variation of splitting caused by c hanging the gate overdrive does not affect the carrier distribution. As discussed above, the current enhancement of strained p channel TG Fin FET s strongly depends on the gate bias if the fin width is large. For narrow width device, its strain enhanced pe rformance can be modeled by (110) planar MOSFET s coefficient and there is no observation of higher coefficient value caused by bulk inversion as in DG FinFET s 3.4 Conclusion Piezoresistance coefficients of MOSFETs with different surface orientations and channel directions are measured under longitudinal, transverse, and biaxial stresses T he results are compared to the bulk Si value from Smith and to other literature publications The extracted coefficients of the (110) MOSFETs are significantly d ifferent from the bulk value, and the surface confinement effect must be taken into account. For the (001) nMOSFETs, the uniaxial coefficients strongly depend on the doping density, electric field, and channel direction. The magnitude of the (001) i s comparable to that of l and is insensitive to the vertical electric field, mak ing biaxial stress promising for nMOSFETs. For the (001) pMOSFETs, the (110) longitudinal uniaxial stress has the highest coefficient which has little dependence on the vertical electric field, and therefore is the most advantageous stress for pMOSFETs. PAGE 56 56 The longitudinal coefficient s of n and p channel TG Fin FETs with fin width down to 20nm were measured and compared to the planar MOSFET and DG FinFET values. Within th e entire fin width range, the strain induced current enhancement for n channel TG Fin FET can be modeled by a linear combination bet ween the ( 001 ) and (110) planar coefficients. For p channel TG Fin FET, the coefficient of narrow width device matches th e theoretical prediction, whereas the coefficient for wide fins is smaller than the ( 001 ) planar MOSET value due to a high carrier concentration in the undoped channel. Both experiment and theoretical results prove that longitudinal stress is beneficial for TG Fin FET current transport. PAGE 57 57 CHAPTER 4 EFFECT OF STRESS ON GAN HEMT RESISTANCE 4.1 Introduction AlGaN/GaN high electron mobility transistors (HEMT) have great potential for high voltage switching and broad band power applications [90 91] owing to the large band gap of GaN [99] Despite the be impaired by the inverse piezoelectr ic effect [50 52] which generates unexpected mechanical stress while the device is operated under high voltage. To clarify the stress effect on HEMT reliability, the stress altered channel resistance has been studied which is a good indicator on reliability issues such as hot electron injection [100] Gauge factors have been extracted experimentall y in previous works [43 48] though the results were confusing due to their large variation from 4 to 40,000. Koehler et al. [49] recently reported a gauge factor of 2.40.5 and attributed the large variation in liter ature to the existence of trapping effects during measurement. To better understand the stress effect on HEMT device channel resistance and determine a reasonable gauge factor, a simulation model is developed in this chapter by considering the stress alte red two dimensional electron gas (2DEG) sheet carrier density and electron mobility. For the first time, uniaxial stress is incorporated into an sp 3 d 5 empirical tight binding model to investigate the stress effect on electron effective mass in wurtzite Ga N 4.2 Theory and Modeling The channel resistance of the AlGaN/GaN HEMT device is inversely proportional to the 2DEG sheet carrier density ( n S ) and channel electron mobility ( e ). This section [2010] American Institute of Physics Chapter 4 is reprinted with permission from [M. Chu A.D.Koehler, A.Gupta, T.Nishida, S.E.Thompson Simulation of AlGaN/GaN high electron mobility transistor gaug e factor based on two dimensional electron gas density and electron mobility J. Appl. Phys., vol.108, pp.104502, November 2010] PAGE 58 58 describes the model and procedure that are used to study the stress effect on n S and e Simulation uncertainty is investigated to ensure more accurate result s 4.2.1 Stress D ependence of 2DEG S heet C arrier D ensit y The 2DEG forms at the interface of the Al GaN and GaN layers, arising from the total polarization difference between them [101] There are two types of polarization: spontaneous polarization ( P SP ) and piezoelectric polarization ( P PE ) Spontaneous polarization exists in both AlGaN and GaN layers since their ratio differs from the ideal wurtzite crystal value ( ) Piezoelectric polarization arises from the piezoelectric effect, which is proportional to the strain. AlGaN/GaN HEMT structure, a thin layer of strained AlGaN due to lattic e mismatch is on top of a thick layer of relax ed GaN. As a result, piezoelectric polarization exists only in the AlGaN layer without external stress. F igure 4 1 A schematically shows the total F rom the total p olarization difference between the AlGaN and GaN layers, t he 2DEG sheet carrier density can be calculated using Eq. ( 4 1 ) [101] (4 1) where x is the Al content, is the total polarization difference between AlGaN and GaN layers, is the dielectric constant, d is the depth of the AlGaN layer, q b is the Schottky Barrier of a gate contact, E F is the Fermi level with respect to the GaN conduction band edge energy, and E C is the conduction band offset at the AlGaN/GaN interface. This work uses an Al content of 0.26. PAGE 59 59 A B Figure 4 1. Illustrations of polarizations in the AlGaN and GaN layers. A) For the as fabricated GaN HEMTs B) For the GaN HEMTs under mechanical bending stress. The mechanically applied stress generat es additional piezoelectric polarization of similar magnitude in both the AlGaN and GaN layers. [2010] American I nstitute of Physics reprint with permission from [M. Chu et al., Simulation of AlGaN/GaN high electron mobility transistor gauge factor based on two dimensional electron gas density and electron mobility J. Appl. Phys., vol.108, pp.104502, Fig.3, Novemb er 2010]. External mechanical stress affects the 2DEG density by generating additional piezoelectric polarization along the [0001] direction. Spontaneous polarization stays the same since it is a n intrinsic material quality. When external stress is applied, additional piezoelectric polarization arises in both layers as shown in F ig ure 4 1 B The amount is proportional to the strain and piezoelectric coeff icients ( ) In this work, the AlGaN and GaN layers are assumed to have the same level of strain due to the fact PAGE 60 60 that in most mechanical bending experiment both layers of the AlGaN/GaN HEMT are significantly thinner than the substrate and therefore are located near the top surface of the wafer. In addition, the piezoelectri c coefficients of AlGaN and GaN are similar. As a result, the difference between the strain induced piezoelectric polarizations of these two layers is close to zero. Therefore, we expect that external stress has little effect on the 2DEG sheet carrier de nsity. 4.2.2 Stress D ependence of C hannel E lectron M obility It has been concluded in Chapter 2 that the mechanical stress alters the channel electron mobility mainly through band warping induced effective mass change. We use a n sp 3 d 5 empirical tight binding method developed in [73] to calculate the GaN band structure and electron effective mass T he unit cell of wurt zite GaN contains four atoms, two anions (N) and two cations (Ga) as shown in F ig ure 4 2 A All nearest neighbor s, p, and d interactions, as well as second nearest neighbor s and p interactions are included using two center approximation [102] which results in a 2626 Ham iltonian matrix. The tight binding parameters used in this work, including five on site one center, eight nearest neighbor two center, and eight second nearest neighbor two center integrals, are list ed in Table IV of [73] The GaN b and structure is obtained by solving the eigenvalue of the Hamiltonian matrix. T he electron effective mass is then calculated along the M, K, and A directions in the reciprocal lattice as shown in F ig ure 4 2 B The effect of mechanical stress is incorporated into the tight binding model by considering the strain induced change in atom location, which results in varied bond PAGE 61 61 length, bond angle, and reciprocal lattice. In this work, in p lane biaxial stress and uniaxial stress along the x and y axis as shown in F igure 4 2 A are considered. A B Figure 4 2. Wurtzite GaN structures in the real and reciprocal spaces. A) U nit cell with 4 basis atoms, where 1 and 3 are Ga atoms, 2 and 4 are N atoms B) The reciprocal lattice for unstrained wurtzite GaN. Electron effective mass is calculated along K, M and A directions. [2010] American Institute of Physics reprint with permission from [M. Chu e t al., Simulation of AlGaN/GaN high electron mobility transistor gauge factor based on two dimensional electron gas density and electron mobility J. Appl. Phys., vol.108, pp.104502, Fig.5, November 2010]. Under biaxial stress, the hexagon in the c plane does not deform but only varies in size as sho wn in F ig ure 4 3 A The resulting strain in all three directions is related to the applied stress through Eq. ( 4 2 ) and ( 4 3 ) (4 2) (4 3) PAGE 62 62 where is the biaxial stress, and C 11 C 12 C 13 and C 33 refer to the GaN elastic stiffness constants. T he primitive translation vectors i n rectangular coordinates becomes and where a is the length of a hexagon side and c is the repeat distance in the z direction. The corresponding reciprocal la ttice vectors are and The reciprocal lattice, under biaxial stress, remains a hexagonal shape. The basis vectors where the atoms are locate d are and The strain varied bond length and bond angle b etween all nearest and second nearest neighbors can then be calculated base d on the new atom location s Since the atom ic layout in the c plane remains hexagonal, the expressions of elements in the 2626 Hamiltonian matrix do not change. The stress depend ent band structure calculation is straight forward, with the bond length and bond angle the only parameters needed to be changed. Under uniaxial stress, however, the in plane hexagonal shape is deformed as shown in F ig ure 4 3B The resulting strain in all three directions is relat ed to the applied stress through Eq. ( 4 4 ) ( 4 5 ) and ( 4 6 ) (4 4) (4 5) (4 6) PAGE 63 63 A B Figure 4 3 Lattice projection on the c plane of GaN under externally applied mechanical stress. A) Under biaxial stress, t he atomic layout remains a hexagonal shape. B) Under uniaxial stress, t he hexagon deforms, and the strain in the x direction differs from the s train in the y direction. [2010] American Institute of Physics reprint with permission from [M. Chu et al., Simulation of AlGaN/GaN high electron mobility transistor gauge factor based on two dimensional electron gas density and electron mobility J. Appl. Phys., vol.108, pp.104502, Fig.6, November 2010] The primitive translation vectors and basis vectors in rectangular coordinates are calculated the same way as under biaxial stress Figure 4 3B shows that under uniaxial stress, t he bond length betwee n the cent ral atom to atom 1 differs from the bond length between the cent ral atom to atom 2 or atom 3 The original symmetry is broken, and t herefore the expressions of Hamiltonian matrix elements derived in [73] need to be re generated, following a standard tight binding method with the two center approximation PAGE 64 64 as describe d in [102] All of atom 1, 2 and 3, as shown in F igure 4 3B are still considere d to be the nearest neighbors to the cent ral atom. During our derivation, it is found that besides the bond length and bond angle, the expressions of g 0 g 1 g 2 and g 3 in [73] also change. The reciprocal lattice, under uniaxial s tress, is not hexagonal in shape. The e ffective mass should be ca lculated along directions bas ed on the new reciprocal lattice vectors. Stress induced band warp ing is determined by, first, whether the applied stress is along a high symmet ric direction and s econd, whether there is strong subband interaction between the conduction band and any other subband. In this work we study the effect of biaxial stress and uniaxial stress along and direction s Although these types of stress es slightly alter the wurtzite crystal symmetry, we expect little change in electron effective mass and thus electron mobility Thi s is because GaN has a large bandgap, and has no energetically adjacent conduction bands. 4.2.3 Simulation U ncertainty In this work, the simulation uncertainty is investigated by considering various elastic stiffness constants and piezoelectric coefficient s of GaN and AlN list ed in literature. The uncertainty arising from the tight binding parameters is neglected, since these parameters are achieved from an empirical tight binding method that is considered to be accurate. To sum up this section, we expect that both the 2DEG sheet carrier density and the electron mobility have a weak dependence on external mechanical stress, leading to a small gauge factor for the AlGaN/GaN HEMT. PAGE 65 65 4.3 Results and Discussion Fig ure 4 4 shows the percentage change of 2DEG sheet carrier density versus tensile stress. A n increase in n S ranging from 0.09 % to 1 .4 % for 500MPa can be achieved depen used in the simulation. This small enhancement matches our expectation due to the fact that th e additional piezoelectric polarization in AlGaN and GaN layers mostly cancel each other out. Figure 4 4 Percent change of 2DEG sheet carrier density under uniaxial tension. Simulation uncertainty is shown as the shaded area by considering various elastic stiffness constants and piezoelectric coefficients. [2010] American Institute of Physics reprint with permission from [M. Chu A.D.Koehler, A.Gupta, T.Nishida, S.E.Thompson Simulation of AlGaN/GaN high electron mobility transistor gaug e factor based on two dimensional electron gas density and electron mobility J. Appl. Phys., vol.108, pp.104502, Fig.4, November 2010] The stress effect on the electron effective mass along the longitudinal, transverse, and out of plane directions are ca lculated and the results are plotted in F ig ure s 4 5 A C respectively. Here we consider the direction to be the channel direction. Therefore, the l ongitudinal t ransverse and o ut of plane directions refer to the PAGE 66 66 and [0001] direction respectively The shaded areas in the plots include all possible simulation results considering simulation uncertainties. Without stress, the longitudinal, transverse, and out of plane effective masses are 0.198m 0 0.197m 0 and 0.189m 0 respectively, whi ch agree with the results in [73] For longitudinal effective mass, biaxial stress has a slightly larger effect than uniaxial stress (~ 3 %/ 5 00MPa under biaxial stress comparing to ~ 1.5 %/ 5 00MPa under longitudinal stress). Similarly, transverse and out of plane effective mass also have a larger change under biaxial stress. For all types of stress, however, the changes in electron effective mass of GaN are much smaller than those of Si (~15 %/ 5 00MPa under <110> uniaxial stress [17] ). This is because the s ubband splitting which is an important factor affecting electron effective mass in Si, does not exist in Ga N. Si is an indirect bandgap material with the conduction band minimum locate d near the X point, leading to six degenerate conduction bands. When stress is applied, the 6 conduction bands split into 2 and 4 valleys, and the splitting cause s electrons to repopulate. Since the conductivity effective mass of these sub bands are different (0.415m 0 for 4 valley comparing to 0.19m 0 for 2 valley), the total average effective mass changes depending on the amount of band splitting. The absence of band splitting and carrier repopulation in GaN causes the electron effective mass and thus mobility to only depend on conduction band warping, which is proved to be small th r ough tight binding calculation s Combining the variation of 2DEG sheet carrier density and electron mobility, the stress induced change in channel resistance o f an AlGaN/GaN HEMT is plotted in F ig ure 4 6 Simulation uncertainties are included. The simulation result is compared to the experimental result presented in [49] in which repeatable gauge factors were PAGE 67 67 obtained after eliminating parasitic charge trapping effects. The simulated gauge factor is determined to be 7.9 5.2, compared to 2.5 0.4 in [49] while the values published in literature rang e from 4 to 40,000 as list ed in T able 4 1 The wide range of gauge factor s list ed in literature is considered to be a result of the trapping effect occurring over the elapsed time of measurements. Our result also agrees with the gauge factors of bulk GaN reported in [44, 103 104] with little variation ( 1 to 3.6) that indicates negligible trapping e ffect. Comparing the simulation result with the experimental result of [49] the best fit set of elastic stiffness constants and piezoelectric coefficients used in simu lation were determined to be C ij (GaN) [105] C ij (AlN) [106] e ij (GaN) [103] and e ij (AlN) [107] as list in T able 4 2 and Table 4 3 4.4 Conclusion Stress was incorporated into a sp 3 d 5 sp 3 empirical ti ght binding method by recalculating the atom locations, the reciprocal lattice, and consequently the bond length, bond angle and the Hamiltonian matrix elements. The tight binding calculation results indicate small change in electron effective mass and th us mobility. External mechanical stress generates additional piezoelectric polarization in both AlGaN and GaN layers that cancel with each other. Therefore, stress has little impact on the 2DEG sheet carrier density. Combining the stress varied 2DEG sheet carrier density and electron mobility, the gauge factor of AlGaN/GaN HEMT d evice was calculated to be 7.9 5.2. This indicates a small stress dependence on the HEMT device channel resistance. The best fit set of elastic stiffness constants and piez oelectric coefficients of GaN and AlN was determined by comparing the simulated and measured gauge factors. PAGE 68 68 A Figure 4 5 C hange of electron effective mass under external ly applied mechanical stress A) Longitudinal stress. B) Transverse stress. C) Bi axial stress. [2010] American Institute of Physics reprint with permission from [M. Chu et al. Simulation of AlGaN/GaN high electron mobility transistor gauge factor based on two dimensional electron gas density and electron mobility J. Appl. Phys., vol.108, pp.104502, Fig.7 (a) November 2010] PAGE 69 69 B C Figure 4 5 Continued. PAGE 70 70 Figure 4 6 C hange of GaN HEMT resistance (R TOT ) under longitudinal stress Symbols represent experimental change in R TOT [49] with uniaxial stress. [2010] American Institute of Physics reprint with permission from [M. Chu et al. Simulation of AlGaN/GaN high electron mobility transistor gauge factor based on two dimensional electron gas density and electron mobility J. Appl. Phys., vol.108, pp.104502, Fig.8, November 2010] PAGE 71 71 Table 4 1. Gauge factors of GaN HEMT s and bulk GaN published in literature. Ref. GF (%) (MPa) Method of Stressing GaN HEMT [49] 2.6 0.114 360 4 point bending [43] 4 0.03 95 3 point bending [44] 42 0.0 05 1 5 3 point bending [108] 75 0.0 4 1 2 6 3 point bending [47] 90 0.14 442 3 point bending [45] 1,259 1.35x10 4 0.42 Cantilever [46] 38,889 3.85x10 4 1.2 Circular Membrane Bulk GaN [44] 3.5 0.005 1 5 3 point bending [103] 1~4 4.25x10 4 1.34 3 point bending [104] 3.6 3 point bending Table 4 2. The best fit set of stiffness constants of GaN and AlN [2010] American Institute of Physics reprint with permission from [M. Chu et al. Simulation of AlGaN/GaN high electron mobility transistor gauge factor based on two dimensional electron gas density and electron mobility J. Appl. Phys., vol.108, pp.104502, Table I, November 2010] C 11 C 12 C 13 C 33 C 44 C 66 GaN 37.3 14.1 8.0 38.7 9.4 11.8 AlN 41 14 10 39 12 11.8 Table 4 3. The best fit set of piezoelectric coefficients of GaN and AlN [2010] American Institute of Physics reprint with permission from [M. Chu et al., Simulation of AlGaN/GaN high electron mobility transistor gauge factor based on two dimensional electron gas density and electr on mobility J. Appl. Phys., vol.108, pp.104502, Table II, November 2010] e 13 e 33 e 15 GaN 0.22 0.22 0.44 AlN 0.48 0.58 1.55 PAGE 72 72 CHAPTER 5 EFFECT OF STRESS ON GAN HEMT GATE LEAKAGE 5.1 Motivation Great potential for high power wide band application has been demonstrated by the AlGaN/GaN HEMT, though its overall performance is limited by reliability issues [100, 109 ] The degradation phenomena of HEMT devices, such as current collapse, V T shift, gate lag, drain lag transconductance fr equency dispersion and barrier breakdown, have been attri buted to either hot electr on or the i nverse piezoelectric effects [100] The degradation mechanisms include both electronic mechanism s as well as mechanical stress. In the theory of the hot electron effect th e high electric field present in the GaN channel enables electrons to gain enough energy to overcome the potential barrier and enter the AlGaN where they are trapped by the donor l ike traps These traps may occur at the AlGaN surface, in the AlGaN bulk region, as well as at the AlGaN/GaN interface. The trapped negative charge acts as a virtual gate [110] which decrease s the channel carrier density and consequen tly results in a decrease of drain current and a possible increase of threshold voltage. I n the theory of inverse piezoelectric effects, additional mechanical strain is generated in the AlGaN layer when an electric bias is applied This mechanical strai n in addition to the lattice mismatch induced strain in the AlGaN layer can exceed the critical mechanical yield stress limit of AlGaN This fracture is postulated to result in a sudden device degradation inducing a much larger gate current [51] To validate this hypothesis, a mechanical bending experiment has been done on GaN HEMTs under high reverse gate bias with zero potential drop between the source and drain [52] An increase of gate current and a decrease of the critical breakdown voltage with tensile stress were observed. PAGE 73 73 As important as researching the catastrophic degradation mechanisms of HEMTs is investigating the pre degradation device behavior such as the reverse biased gate leakage current can provide fundamental understandin g of the physics behind gradual degradation. It has been proposed that the defects in the bulk AlGaN barrier, at the AlGaN surface, or at the AlGaN/GaN interface are likely to contribute to the gate leakage current [111 112] During device o peration, the AlGaN layer has additional strain induced by the inverse pi ezoelectric effect, which may shift the defect energy levels. This change i n defect characteristics is expected to affect the gate leakage current and device reliability. T o better predict GaN HEMT performance and reliability, it is important to understand the effects of mechanical strain on the HEMT gate leakage current. In this chapter, a comprehensive study of the gate leakage mechanisms in the reverse biased GaN HEMT is presented. Mechanical stress is incorporated into the model in order to simulate the stress altered gate leakage current. 5.2 Gate Leakage Mechanisms I n this section, several potential leakage mechanisms in the reverse biased GaN HEMT are modeled, as a function of electric field in the AlGaN barrier and temperature B y comparing with the experiment ally observed gate current density, the dominant gate leakage mechanism is determined. T he effects of external stress on the electron out of plane effective mass, AlGaN electric field, and defect energy level s are investigated. Comb ining these factors, the stress sensitivity of the gate current at various electric field strengths and temperatures is modeled and analyzed PAGE 74 74 5.2. 1 Literature R eview S everal leakage mechanisms have been proposed in the literature to explain the GaN HEMT gate current at various temperature and field strength conditions. Zhang et al. [113] noted that direct tunneling of electrons from the metal gate into the GaN bulk dominates the gate current measured at low temperature (<130K), while the gate current measured at room temperature or a bove follows the Poole Frenkel E mission (PFE) trend. Mi trofanov et al. [114] and Yan et al. [115] also brought up a similar conclusion that PFE dominates the room temperature gate leakage. Karmalkar et al. [111] develop ed a bulk defect assisted tunneling gate leakage model In their model, the electrons undergo two thermal assisted direct tunneling processes : from the gate to the trap and from the trap to the bulk GaN. They were able to reproduce the ir experiment al results on various devices by using different sets of parameters including the defect level, defect density, and the Schottky barrier height. Sathaiya et al. [112] proposed a similar two step thermal assisted tunneling model to be the dominant leakage mechanism i n the GaN HEMT. D espite many models, only a few of them [115] considered the backward current which is essential in order to balance the forward current to properly satisfy the zero net current at zero bias equilibrium condition S ummarizing the literature findings, a single leakage mecha nism dominating the GaN HEMT gate current at all conditions does not exist. T he dominant leakage mechanism d epend s on temperature, gate bias, and device quality. I n the following, several candidate leakage mechanisms are examined before the dominant one is determined for our devices under our experiment al conditions T he backward current will be included in our model to study the stress dependence of the gate leakage current. PAGE 75 75 A B Figure 5 1. Schematic illustration s o f the gate leakage process in a reverse biased GaN HEMT. A) Fowler Nordheim tunneling B) T hermionic F ield E mission. Figure 5 2. Relation between the gate bias and the vertical electric field in the 18nm thick Al 0.26 Ga 0.74 N layer. Th is relation has been confirmed by both TCAD simulation and experimental measurements. 5.2. 2 Direct T unneling I n this work, the direct tunneling takes into account the Fowler Nordheim tunneling (FN) and Thermioni c Field Emission (TFE). Figure 5 1 schematically illustrates the PAGE 76 76 direct tun neling process es To simulate this leakage current, t he classical models for FN and TFE are used. These models can be expressed analytical ly as Eq. (5 1) for the FN tunneling [116] and Eq. ( 5 2 ) through ( 5 4 ) for the TFE [111] respectively. (5 1) (5 2) (5 3) (5 4) H ere, q m e h k and A* are the electron charge, the free electron mass, the Planck constant, the Boltzmann constant, and th e effective Richardson constant respectively E is the electric field strength in the AlGaN layer. m is the out of plane effective mass of electrons inside AlGaN. f FD is the Fermi Dirac distribution funct ion of electrons on the metal side. B is the Schottky barrier height between the metal gate and the AlGaN, obtained from Eq. ( 5 5 ) considering the barrier lowering due to image force and band gap narrowing [111] T he values of B0 1 and T are 1.4eV, 0.4 and 2.4 10 4 V/k for AlGaN [111] (5 5) I n order to compare simulation and experimental results, an accurate transformation from the applied gate voltage to the vertical electric field in the AlGaN PAGE 77 77 barrier is required. For this purpose, a Sentaurus TCAD model is developed and the electric field at various bias conditions is simulated. In addition, high frequency (capacitance voltage) CV measurement and threshold voltage measurement are carried out to confirm the voltage field relation [117] The result is shown in Figure 5 2. It ind icates that the AlGaN electric field under the gate region begins to saturate at Vg=1.6V. Beyond this threshold voltage, the 2DEG is depleted and only the electric field at the edge of the gate increases further. To avoid ambiguity in future discussion we defined above threshold as the condition at which the 2DEG exists and the device is on. The below threshold is defined as the condition at which the 2DEG is completely depleted and the device is off. Figure 5 3. Simulation results of the FN tu nneling current and the TFE current at room temperature. PAGE 78 78 Figure 5 3 shows the simulated direct tunneling current and the experiment al measured gate current at room temperature. The result shows that the direct tunneling currents are several orders smaller than the experiment al observation until at least 2.5MV/cm. This is expected since the AlGaN layer (18nm) is too thick for the electrons to have a high tunneling rate. In addi tion, the Schottky barrier height (~1.4eV) is unlikely to result in a high thermal emission rate at room temperature. A s a result, direct tunneling is not the dominant leakage mechanism in our devices when bia sed above the threshold (Vg> 1.6 V). At below threshold, there can be a direct tunneling path at the edge of the gate once the edge electric field is high enough. 5.2. 3 Bulk Trap Assisted L eakage Figure 5 4. Schematic illustration of the 2 step bulk trap assisted leakage process. T his work adopts F leischer s two step leakage model [118] and develops it to incorporate the thermal contribution in step 2. In the first step of our model, the electrons tunnel from the metal gate to the trap level through direct tunnel ing or thermal PAGE 79 79 assisted direct tunneling. I n step 2, electrons escape from the trap into bulk GaN through direct tunneling, Poole Frenkel emission [119] or Phonon assisted tunneling [120 121] The complete leakage process is shown schematically in Figure 5 4 The tun neling (emission) rates of step 1 and 2 are calculated from Eq. ( 5 6 ) and ( 5 7 ) (5 6) (5 7) Here, N t is the bulk defect density f is the probability that the defect l evel is filled with an electron, P 1 and P 2 are the t unneling (emission) probability, and C t is given by (5 8) where E T is the defect level and E 1 is the total energy of an electron (0.2eV [118] ). P 1 is calculated following the standard direct tunneling expression (5 9) a nd P 2 is calculated using either the FN tunneling model, the classical Poole Frenkel emi ssion model or the phonon assisted tunneling model developed by Pipinys et al. [120] A t steady state, R 1 should equal R 2 After a straight forward mathematical derivation, the tunneling rate R is obtained (5 10) T he overall gate current density is then calculated by integrating R over the thickness of the entire AlGaN barrier. (5 11) PAGE 80 80 Figure 5 5. Simulation result of the 2 step bulk trap assisted leakage current at room temperature. I n the current model, a uniform defect distribution is assumed across the AlGaN barrier. The AlGaN potential profile is assumed to be triangular, which is reasonable for devices biased above threshold. We also considered only one single effective defect level as a fitting parameter. Figure 5 5 shows the calculated and experiment al gate current density at room temperature, with N t =5e1 7 /cm 3 and E T =0.82 eV. T he value of E T significantly affects the resulting current magnitude. In fact, E T has opposite effects on P 1 and P 2 With small E T the energy difference between the metal gate Fermi level and the defect level is large, and the fi rst step of electrons tunneling from the metal gate into the defect level is unlikely to happen. W ith large E T the energy difference between the defect level and the AlGaN conduction band is large, and the step two probabilit y of electrons PAGE 81 81 undergoing PFE or PAT becomes smaller. T he value of E T =0.82 eV wa s obtained by trying to match the two step model and the experiment al results within the same order of magnitude under a reasonable defect density N t =5e17 /cm 3 Under this condition, it is found that the calculated current density does not depend on the chosen mechanism for the P 2 step. This indicates that the first step of electrons tunneling from the metal gate into the defect level limits the bulk trap assisted leakage current. Regardless of what leakage m echanism for step two and fitting parameters are chosen, a good match between the model and experiment al results is unachievable. This suggests that the bul k trap assisted tunneling mechanism is not the dominant leakage mechani sm in our GaN HEMT devices. 5.2. 4 Poole Frenkel E mission from Surface S tates I t has been proposed in literature that the donor like surface states can trap hot electrons and act as a virtual gate Here these surface states are considered to participate in the gate leakage process through Poole Frenkel E mission in the reverse biased GaN HEMT. This work develops a surface state related leakage mechanism, as s hown schematically in Figure 5 6 J PF stands for the forward Poole Frenkel E mission current and J back stands for a backward current. Since a non zero electric field exists at zero gate bias leading to a non zero forward current, it is essential to include the J back in the model to ensure zero net current at zero applied gate bias at equilibrium At any bias, the net gate current is obtained from the difference between the forward and backward currents, J PF J back T he forward current is modeled by a modified Poole Frenkel Emission mechanism. T he classical Poole Frenkel equation has the form [119] PAGE 82 82 (5 12) where represents the defect barrier lowering due to the electric field in the AlGaN layer. C is a constant related to the electron mobility and interface defect density [122] It is important to note that it is the high frequency permittivity that should be used as the AlGaN permittivity [123] in the barrier lowering term. From and the calculated permittivity for Al 0.26 Ga 0.74 N is 5.1. Other parameters have their usual meanings. Figure 5 6. Schematic illustration of the Poole Frenkel Emission process from the surface states. The net gate current is the difference between the forward Poole Frenkel current and the backward electron diffusion current. PAGE 83 83 However, it is well know n that the slope of a classical Poole Frenkel model Log(J/E) vs. E 1/2 often deviates from the experiment al observations [124] A more generalized Poole Frenkel E mission model has been proposed as [123] (5 13) Here, the parameter r accounts for compensation effects between the donor like defect states and any possible acceptor states. It equals 1 for full compensation and 2 for no compensation. Depending on how heavy the compensation is, r varies between 1 and 2. In this work, the r parameter is treated as a fitting parameter to best fit our experiment al observations. Figure 5 7. Modeling results of the Poole Frenkel Emission from surface states at various temperatures. The experimental results are also shown for comparison. The inset shows the J vs. E in the linear scale. PAGE 84 84 F or the backward current, its physical mechanism is still unclear. It possibly relates to the bulk defect assisted electron hopping from the AlGaN/GaN interface back to the AlGaN surface states as shown in Figure 5 6 It is likely to have a diffusion type of behavior due to the density gradient of the trapped electrons inside the AlGaN. F or the modeling purpose, we adopt the form of tunneling current proposed by Yan [115] Instead of using the classical direct tunneling model, we incorporated a parameter C in the pre exponential term of Eq. ( 5 14 ) which satisfies J back = J PF at Vg=0V. Since J PF increases significantly once Vg 0 and J back becomes negligible, estimating J back from a rough model should not affect the overall calculation (5 14) C ombining the forw ard and backward current, the simulated Log(J/E) vs. E 1/2 at various temperatures is shown in Figure 5 7, together with the experimentally measured results for comparison. The inset shows the J vs. E in the linear scale. Good agreement between the model calculation and experiment is achieved by using the set of fitting parameters as noted in Table 5 1. The uncertainties of the fitting parameters are also listed in Table 5 1. T he r parameter determin es the slope of the Poole Frenkel plot and its uncertainty is calculated based on a 10% slope variation. E T determines the temperature dependence of the modeled gate leakage current I ts uncertainty is determined at the moderate electric field (Poole Fren kel dominated ) region, from the condition that the simulated temperature dependence has less than 10% deviation from the experimental observation. The variations of r and E T alter the value of C as shown in Table 5 1. T he value of the r parameter (1.25) in dicates that a certain level of PAGE 85 85 compensation exists. I t is also interesting to note that the fitting parameter E T =0.49 closely agrees with our experimentally extracted E T =0.5. B ased on the above discussion, it is concluded that the dominant gate leakage mechanism in our GaN HEMTs is the Poole Frenkel Emission from the surface states. Table 5 1. The Poole Frenkel Emission fitting parameters. r E T C 1.2 5 ( 0.06) 0.49 ( 0.04) 7.3 10 6 ( 0.22 10 6 ~ 25 10 6 ) 5.3 Effects of Stress on Gate Leakage I n this section, external stress is incorporated into the previously developed gate leakage model to study the stress altered gate current. In the model of Poole Frenkel Emission from surface states stress can affect the gate current through the electron out of plane effective mass, the AlGaN electric field, the defect energy level, and the r parameter. 5.3.1 S tress Dependent Parameters W e used the sp 3 d 5 sp 3 tight binding model developed in Chapter 4 to calculate the change of electron out of plane effective mass under both uniaxial and biaxial stress. In the experiment, the uniaxial stress is induced by wafer bending, and the biaxial stress is induced by th e inverse piezoelectric effect when gate bias is present. Figure 5 8 shows the tight binding calculation results for both uniaxial and biaxial stress. I t is observed that the electron out of plane effective mass has negligible change under both stresses (<0.5% per 400MPa). S tress causes variation in the AlGaN polarization due to the piezoelectric effect, leading to a change in the electric field strength. Following the discussion in Chapter 4, this additional piezoelectric polarization can be calculated by where is PAGE 86 86 the piezoelectric constants and is the strain. T he simulation results for both uniaxial and biaxial stress are shown in Figure 5 9. Approximately 0.7% and 1.4% change in polarization is predicted for 400MPa uniaxial and biaxial stress, respectively. This observation indicates a weak stress dependence of the AlGaN electric field. Figure 5 8. Change of the electron out of plane effective mass under uniaxial and biaxial stress. Stress has been proposed to shift the defect energy level by changing the atom defect bond angle and bond length [125] It was observed that both tensile and compressive stresses decreases Pb 1 (or Pb 2 ) defect levels for Si MOSFETs, resulting i n an increase of gate leakage current [125] For GaN HEMTs, our experimental results in Figure 5 10 show that tensile stress increases gate current, while compressive stress decreases gate current. This can be explained qualitatively with Figure 5 11, taking the N vacancy as an example. As mentioned in Chapter 4, approximately 2.8GPa biaxial stress is present in the AlGaN layer due to lattice mismatch. Compared to the PAGE 87 87 unstrained lattice, the bond angle 1 decreases and 2 increases, and the bond length L 1 decreases and L 2 increases. The defect in the AlGaN becomes less stable under this large stress, and its energy level shifts up towards the conduction band. When a tensile wafer bending stress is applied, a similar trend of bond angle and bond length va riation is expected. I t adds to the effect of biaxial stress and shifts the defect level further up. In contrast, a compressive wafer bending stress tends to relax the strain from the lattice mismatch, therefore results in a downward shift of the defect level towards its original state. Since the gate current exponentially depends on the defect level, a decrease (or increase) of the defect level from tensile (or compressive) stress increases (or decreases) the gate current. I n this work, E T is treated as a fitting parameter between simulation and experiment. A more rigorous theoretical model is yet to be developed for a quantitative prediction of the stress altered defect level. Figure 5 9. Change of polarization in the AlGaN layer under uniaxial and biaxial stress. PAGE 88 88 Figure 5 10. Experimentally measured gate leakage current change under uniaxial tensile and compressive bending stress. The current is measured under V G = 0.25V. T he r parameter also varies with stress, due to the shift of the defect level as shown in Figure 5 12 U nder tensile stress, the defect level shifts towards the conduction band, resulting in a higher probability of electron emission from the defect state in to the conduction band. This indicates weaker compensation and thus a higher r value. In contrast when compressive stress is applied, the defect level shifts downwards, resulting in a smaller probability of emission This indicates stronger compensation and thus a smaller r value. PAGE 89 89 5.3.2 R esults and Discussions Incorporat ing the above discussed factors into the model of Poole Frenkel Emission from surface states, the change in the GaN HEMT gate current under uniaxial bending stress is calculated and shown in Figure 5 13. At room temperature, a close match between the simu lation and experimental results at various electric fields, E AlGaN is obtained by fitting E T =1.6 0.3 meV/GPa and r =0.028 0.004 /GPa. The uncertainties of E T and r are determined based on the condition that the modeled gate current change has less than 10% deviation from the experimental results including error bars. The smaller stress sensitivity of the normalized gate leakage current change in Figure 5 13 indicate s a weaker stress dependence at higher reverse gate bias. It also shows that the stress dependence of the gate current decreases with increasing temperature. The Poole Frenkel Emission barrier is lowered at higher gate bias, and the electron thermal ener gy increases at higher temperature. Both effects indicate a stronger electron emission and weaker compensation. And therefore, the stress altered r parameter (representing compensation) has smaller effect compared to the low bias (or lower temperature) c ase where the compensation effect is more important. 5.4 Conclusion In this chapter, several possible gate leakage mechanisms have been examined and the dominant leakage in the reverse biased GaN HEMT (above threshold) is determined to be the Poole Frenke l Emission from AlGaN surface states. Its stress dependence is investigated through the stress altered electron out of plane effective mass, the AlGaN electric field, the defect level, and the r parameter. It increases PAGE 90 90 (decreases) under tensile (compress ive) stress, with less than 2% change per 100MPa stress. For the GaN HEMT biased below threshold, the electron direct tunneling from the gate edge becomes important, which is expected to have negligible stress dependence. Figure 5 11. Schematic illustr ations of the defect bond angle and bond length variation, and the defect level shift under lattice mismatch and wafer bending stress. Figure 5 12. Schematic illustration of the ef fect of externally applied mechanical stress on the r parameter. PAGE 91 91 Figure 5 13. Simulation results for the stress altered GaN HEMT gate leakage current at various temperatures. The simulation is based on the model of Poole Frenkel Emission from surface states. The room temperature experimental results are also shown. PAGE 92 92 CHAPTER 6 DFT CALCULATION FOR GAN The density functional theory (DFT) calculation is investigated and applied to bulk GaN. T his chapter describes and discusses the DFT calculation procedure to obtain the GaN band structure with a correct band gap. I n or der to investigate the effects of strain on th e GaN HEMT gate leakage current, a DFT strategy for defect level calculation is then explored. 6 .1 DFT Introduction 6. 1.1 B asic C oncept of DFT DFT is a quantum mechanical method for calculating electronic structures of material systems. It is especially popular in the investigation of ground state properties of relatively large systems whose supercell typically contains ~100 atoms. T he key idea behind DFT is to solve the Schrdinger Equations and view the solutions as a functional of electron density, instead of the spatial coordinates of each individual electron [126] T he classical Schrdinger Equation is described as (5 15) The three terms inside the bracket represent the kinetic energy of each electron, the interaction between each electron and the nuclei, and the electron electron interaction. For a system containing N electr ons, the full solution is a function of 3N coordinates. As a result, a rigorous solution of the Schrdinger Equation becomes a formidable task for practical material systems that have more than tens of electrons. In addition, the factors that we a re actually interested in and can be physically measured are not the individual electron wave f unctions, but the possibility that a set of electrons PAGE 93 93 are present at a real space point. T his is directly relate d to the electron density described as By solving the Schrdinger Equation using a functional of the electron density, the solution becomes only 3 dimensional. The den sity functional theory is based on two fundamental mathematical theorems proved by Kohn and Hohenberg [127] (1) The ground state energy from s equation is a unique functional of the electron density (2) The electron density that minimizes the energy of the overall functional is the true electron density corresponding to the full solution of the Schrdinger equation Based on these theorems, there exists a unique ground state electron density corresponding to a specific ground state wave function. As a result, the Schrdinger equation can be solved by f inding the electron density that only depends on three spatial coordinates, rather than the ele ctron wave function, which depends on 3N variables. To solve for the right electron density, Kohn and Sham derived a set of equations in 1965 [128] in which each equation involves only one electron. The Kohn Sham equation is expressed as (5 16) The three potentials inside the bracket represent the electron nuclei interaction, the Hartree potential, and the exchange and correlation potential. The Hartree potential describes the Coulomb repulsion between a single electron and the total electron de nsity. Since the total electron density also has the contribution from the electron itself the unphysical interaction between this electron and itself is included in the V H term. As a result, researchers use the V XC term to correct this self interaction fault Formulating PAGE 94 94 and choosing a physically reasonable V XC theme is of particular importance for a successful description of material system by DFT. D eveloping the exchange and correlation functional remains a hot topic in material and chemistry scienc e. The most popular ones are the Local Density Approximation (LDA) [128 132] Generalized Gradient Approximation (GGA) [133 136] LDA +U method [137 138] and Hybrid Functional method [139 143] Figure 6 1. The self consistent procedure for a standard DFT calculation The Kohn Sham equations must be solved self con sistently [144] as shown in Figure 6 1 A trial electron density defines the Kohn Sham equations to solve for the single electron wave functions. The resulting wave functions are used to calculate a PAGE 95 95 new electron density. If the difference between the initial density and the calculated d ensity is within a pre defined stopping criterion then this result is determined to be the right ground state electron density. T he corresponding energy is the true total energy of the system. If the two densities are too different, then the trial elect ron density will be modified in a pre defined way. This will serve as the new trial electron density and the whole process repeats until the stopping criterion is met 6 .1.2 W hy C hoose DFT T here are several modeling methods capable of calculating the electronic structure of a set of atoms These methods include the ab initio (such as DFT, molecular dynamic), semi empirical (such as semi empirical ps e udopotential), or empirical (such as tight binding and k p method) methods In C hapter 4, the tight b inding method was used to calculate the stress altered GaN band structure and electron effective mass. The calculation process is relatively straight forward and the stress effects can be physically previewed through the change of bond length and bond ang le between neighboring atoms, which results in a change in the atom atom interaction. However, the tight binding calculation requires several input parameters such as the on site one center and nearest neighbor two center integrals. These parameters need to be obtained either from experiment or first principle calculation. There are not available defect related parameters for GaN, and therefore the tight binding method is not able to provide information about the defect electronic structure of GaN. In th is work, the DFT method was chosen to investigate the feasibility of defect energy calculation due to the following reasons: (1) DFT is a first principle calculation. Ideally it does no t depend on any empirical inputs. (2) DFT is computationally more eff icient than the more rigorous first principle methods such as the Hartree Foc k PAGE 96 96 calculation. Thus it is capable of dealing with practical systems that contains hundreds of electrons. (3) It is straight forward to incorporate external stress into the DFT model by manually modifying atom locations. (4) It is also straight forward to put defects in to the system, regardless of what type the defect is and where it occur s (5) There are several available simulation packages (such as SeqQuest, Socorro, and VAS P) to make DFT user friendly. Researchers may perform their own calculations using such simulation packages w ithout full understanding of the derivation, parameterization and computation algorith m for the DFT method The DFT calculation delivers useful information about material systems. It provides the true lattice structure by relaxing the atomic sites until a minimu m energy is obtained. It can also predict what type s of defects are more likely to form by computing the energy difference between syste ms with and without defects. This energy difference is called the defect formation energy The lower the formation energy, the easier it is for this particular type of defect to form The DFT calculation also provides the electronic band structure of the material, from which the defect levels and density of states can be derived. In this work, the defect energy level is the most desired property that is important for GaN HEMT gate leakage simulation. 6 .1.3 DFT Calculation P rocedure U sing VASP In this work, DFT calculation is executed using Vienna Ab initio Simulation Package (VASP) [145 146] VASP is a complex package for performing ab initio quantum mechanical molecular dynamic simulati ons using psudopotentials or the projector augmented wave (PAW) method and a plane wave basis set as def ined in the VASP user manual [147] To ensure reliable calculations, four input files must be explicitly provided with proper information included. These four files are named as PAGE 97 97 POSCAR, POTCAR, KPOINTS, and INCAR. The general DFT calculation in VASP starts up in a procedure as shown in Figure 6 2 Figure 6 2 Preparation of VASP input files. The basic input files are the POSCAR, POTCAR, INCAR, and KPOINTS file. In the POSCAR file, the system lattice geometry is carefully construct ed by defining the basic vectors of the unit cell (or supercell) and the ionic positions. The POSCAR file also provides an optional tag Selective dynamics through which the PAGE 98 98 users have additional freedo m to fix coordinates of certain atoms while allow ing others to relax. This option is especially useful for defect calculation in which only ions near the defect site should relax. It is also useful for calculation of systems with external stress applied. In this case, the boundary condition can be satisfied by fixing certain ions. In the POTCAR file, the proper pseudopotential must be chosen for each element species within the system. VASP uses either the ultra soft pseudopotential ( US PP) or the PAW met hod [148] It was proven that both PAW and US PP give same results within 0.1% for semiconductors [148] The KPOINTS file defines the k mesh in the reciprocal space on which a practical DFT calculation is carried out. The choice of k points in the Brillouin Zone is very important, because in a DFT calculation a large portion of the computati on falls into evaluating the k space integrals. A denser k mesh provides more accurate result s with the penalty of higher computational cost. There is always a trade off between accuracy and efficiency. What researchers usually do is to first test the c onvergence of the number of k points, and choose the smallest number of k points that satisfies the convergence requirement. T he INCAR file defines a large number of important parameters that specify a DFT calculation goal ( to do ) and determine a p articular way to execute the calculation ( how to do ). These include choosing the plane wave cut off energy, the relaxation algorith m, and the calculation stopping criterion W ith the proper POSCAR, POTCAR, KPOINTS, and INCAR files, a DFT calculation can be started. Once the calculation is completed, several output files will PAGE 99 99 be created providing the relaxed lattice geometry, the computed wave function, the electron density, the system total energy, and the computation time These output files can be the final results of a task, or they can be used as complementary input files for a further DFT calculation step to solve more complicate problems. 6 .2 DFT Calculation for Bulk GaN I n this section a detail ed procedure of the DFT calculation for obtaining the GaN band structure with correct band gap is discussed. Since the defect levels inside the GaN band gap is the most essential parameter in a gate leakage simulation, the defect calculation strategy should be determined in a wa y that ensures correct band gap. 6 .2.1 S tandard DFT C alculation The GaN band structure is calculated using the standard DFT method with the GGA defined by the PBE functional [134] The calculation is completed in two steps. In the first step, a self consistent DFT calculation is carried out on a k mesh defined by the Monkhorst and Pack approach [149] The chosen number of k points is 6 6 4 in the k x k y and k z directions, respectively. This set of k points has been proven to have converged DFT results for GaN [150 151] The output wave function file (WAV ECAR) and the relaxed lattice geometry (CONTCAR) from this step will be used as a complementary input file for the next step. In the second step, a non self consistent DFT calculation is carried out on a set of denser k points along a high symmetric direc tion. T he band structure along this direction is then constructed by plotting the resulting eigen energies associate d with each k point. The INCAR, POSCAR and KPOINTS files for unstrained GaN calculations are attached and described in Appendix A, B and C respectively. The strained GaN POSCAR files are obtained by modifying the unstrained basic vectors from [b x b y b z ] to PAGE 100 100 [b x (1+ xx ), b y (1+ yy ), b z (1+ zz )], where xx yy and zz are calculated by Eq. (4 2) through Eq. (4 6). Table 6 1 The standard DFT PBE calculation results for the GaN 4 atom unit cell and 32 atom supercell structures, with and without external stress. U nstrained B iaxial U niaxial <11 0> 4 atom Done (1hr) Done (1 hr) Done (1 hr) 32 atom Done (36 hr) Done (40 hr) Done (48 hr) 4 compares to 32 Same results Same results Same results Calculations based on both a GaN unit cell (2 Ga atoms and 2 N atoms) and a 32 atom supercell (16 Ga atoms and 16 N atoms) are performed, with and without external stress. The DFT tasks are organized in Table 6 1 Using 8 CPUs simultaneously, the 4 atom calculations took about one hour to finish and the 32 atom calculations took up to two days. From the observation that both input geometries give the same results, it is co nfirmed that a POSCAR file containing only a single unit cell is able to properly provide the bulk GaN properties. This conclusion indicates that the DFT strategy determined from a 4 atom bulk calculation can be used in the defect calculation with a much larger supercell structure Figure 6 3 A shows the GaN band structure along the M A direction obtained from DFT calculation with the GGA functional defined by Perdew, Burke, and Ernzerhof (PBE) [134] T he calculated band gap is 1.81eV, which is much smaller than the experiment al observation of 3.4~3.5eV. This is the well known band gap problem in the standard DFT method, causing by the absenc e of the derivative discontinuity in the LDA and GGA functionals [152] In the following sub section, DFT calculations with the +U correction and the hybrid functional supplement are performed for the purpose of overcoming the band gap problem. PAGE 101 101 A B Figure 6 3 The calculated E k diagrams for bulk GaN A) Using the standard DFT PBE functionals. B) Using the LDA+U method. 6 .2.2 B and gap C orrection The LDA +U method The LDA and GGA are orbital independent potentials that treat the semicore states as core states, which tend to over localize the semicore electrons. Therefore, the interaction between the semicore states and the valence band maximum m ay not be properly d escribed [153] which causes error in the valence band maximum and thus the band gap value. In the LDA+U approach, the electrons are classified into two groups. For the localized electrons, an additional Coulomb repulsion term U is introduced into the exchange and correlation potential, while the delocalized electrons are described by the usual orbital independent potential in LDA (GGA). The parameter U is a fitting parameter that reproduces the correct band gap. The modified INCAR file for +U calculation is shown in Appendix A The fitting paramete r U represents the effective on site Coulomb interaction, and the paramete r J represents the effective on site Exchange interaction. T he GaN band s tructure calculated from the LDA +U m ethod is shown in Figure 6 3 B The resulting band gap is 1.83eV. Different choices of U an d J para meters end up with similar band gap. This PAGE 102 102 indicates that the under estimation of bandgap does not result from the LDA (GGA) description of semicore states. H ybrid functionals H ybrid functionals have become increasingly popular in the investigatio n of defects in solids [154 157] T he core concept of hybri d functionals is to mix the LDA ( GGA ) exchange potentials with the non local Hartree Fork exchange potential while keeping the correlatio n pot ential as described by LDA ( GGA ) [152] Several forms of hybrid functionals have been proposed in the literature to describe various material systems. In this work, the hybrid functional developed by Heyd, Scuseria, and Ernzerhof (HSE) is used [143] I n the HSE the short range exchange potential is formed by mixing PBE with the Hartree Foc k potential. The long range exchange potential follows the usual PBE potential. This range separation treatment is instrumental in overc oming the band gap problem by providing a more realistic potential. However, the partial inclusion of the Hartree Fork potential tremendous l y increases the computational cost since it involves four center integrals. Due to its high computational cost, instead of starting from the beginning of a problem, the HSE calculation is always performed after a standard DFT calculation step. O btaining t he band structure by HSE involves three steps: (1) A standard self consistent DFT PBE calculation, (2) A self consistent HSE calculation basing on the WAVECAR and the CONTCAR files from step 1, (3) A non self consistent HSE calculation along a high symmetry direction. Step 1 is the same DFT PBE calcul ation as described before The INCAR files for step 2 and step 3 are shown in Appendix A A 4 atom unit cell is used in the HSE calculation. PAGE 103 103 In the HSE implemented by VASP, a researcher has the freedo m of setting the ratio between the PBE and Hartree Fork exchange potential through the parameter AEXX This ratio directly affects the calculated bandgap value and can be treated as a fitting parameter. It is by default 25% Hartree Fork. Using this de fault ratio, a bandgap of 2.85eV is obtained. By adjusting the ratio to 31%, we can obtain a 3.47eV bandgap. Table 6 2 summarizes the bandgap values calculated by the DFT PBE, LDA+ U default HSE, and adjusted HSE functionals. Figure 6 4 shows the GaN ba nd structure along the M direction calculated by HSE with the ratio of 31%. For the purpose of comparison, the band structures obtained from the tight binding method discussed in Chapter 4 and from the empirical pseudopotential method [158] are also shown. T he calculated HSE bandgap of 3.47eV is in close agreement with the values predicted from the other two methods, as well as the experimental bandgap (3.4~3.5eV). Table 6 2. GaN bandgaps calculated from the DFT PBE, LDA+U, and HSE functionals. DFT PBE LDA+U HSE(default) HSE (AEXX=0.31) Eg 1.81 eV 1.83 eV 2.85 eV 3.47 eV A B C Figure 6 4. The E k diagrams for bulk GaN A) C alculated from the DFT HSE. B) The tight binding method. C) T he e mpirical pseudopotential method PAGE 104 104 I t is important to note that the 4 atom HSE calculation for GaN band structure took approximate ly 3 days to finish by using 8 CPUs simultaneously. T he computational cost is 72 times more than the standard DFT calculation. As a conclusion, the GaN band structure with correct bandgap is obtained by using the HSE me thod with a mixing ratio of 31%. Th e setup and the calculation procedure may be employed in the DFT strategy for defect calculation in GaN. 6 .3 GaN Defect Calculations 6 .3.1 Literature R eview Potential defect levels in GaN HEMT have been extensively studied through various experiment techniques, such as drain current deep level transient spectroscopic (DLTS) [159 165] drain leakage current measurement [166] SIMS characterization [167] and photoionization spectrosco py [168] A large range of defect levels were observed within the GaN band gap region, as shown in Figure 6 5 However, despite many previous experimental studies, the defect na ture and their origin s are still not well understood. R ecently, Puzyrev et al. proposed a GaN HEMT degradation model in which active defects are generated by hot electrons through dehydrogenation of the H passivated pre existing defects [169] They performed first principle DFT calculations on candidate defects (N Ga V Ga V N V Ga etc. ) and concluded that different defects are responsible for the degradation of HEMTs fabricated under different conditions. A s imilar technique can be used in the GaN HEMT gate leakage study. By combining the stressed gate leakage measurement and the DFT cal culation the reverse biased gate leakage mechanism can be better understood. If both the pre stressed defect level and the stress induced level shift match between experiment and PAGE 105 105 simulation the corresponding defect and the related leakage model may be confirmed to be the dominant leakage mechanism in the particular GaN HEMT device Figure 6 5 Literature published defect levels observed in GaN HEMTs The defect level s distribute a wide energy range inside the GaN bandgap. [159 168, 170 171] 6 .3.2 T est Calculation and Computational I ssues A DFT calculation under the pre determined HSE strategy was performed to try to obtain candidate defect levels. The chosen candidate defects are nitrogen vacan cy, nitrogen gall ium divacancy, and oxygen substituting nitrogen, because the se defects are close to the conduction band edge and expected to contribute in Po ole Frenkel Emission F or defect related DFT calculation, a relatively large supercell is commonly used to minimize the interaction between defects in neighboring supercells. For wurtzite GaN, it is recommended to use supercell s with 32, 48, 96 or 128 atoms. In this work, a 32 atom supercell was constructed with only one defect inside. For a supercell calculation, employing a single k point for the Brillouin zone integrations is enough to obtain converged results. PAGE 106 106 By using 8 CPUs simultaneously, t he first step of the standard DFT PBE calculation took on average 5 days to finish, and the second step of the self consistent HSE calculation took a tremendous ly longer time (estimated 24 weeks) to finish. In order to have useful results within a realist ic time, more computational facilities and advanced parallelism algorith ms are needed for the HSE based defect calculations. DFT simulation of stress effect on defect energy levels is therefore a potential topic for future work. 6 .4 Conclusion The DFT method was used to accurately calculate the GaN band structure and band gap. T he detail ed pr ocedure to obtain the correct band gap value was discussed. By using the HSE hybrid functional with a mixing ratio of 31% the GaN band structure with correct bandg ap can be achieved Finally a DFT calculation strategy for the strain effect on GaN defect level was outlined. This effort is suggested as a possible future study. PAGE 107 107 CHAPTER 7 SUMMARY AND RECOMMENDATION S FOR FUTURE WORK 7 .1 Summary This dissertation focused on the piezoresistive properties of field effect transistors with various device structures (planar MOSFETs, TG FinFETs, HEMTs) and vari ous channel materials (Si, GaN), as well as the effects of mechanical stress on GaN HEMT gate leakage current A systematic study of the piezoresist ive properties of Si planar MOSFETs and TG FinFETs has been reported. Mechanical stress wa s applied using four point and concentric ring wafer bending setup s I t wa s found that the piezoresistive properties of most FE Ts vary from the strained characteristics of bulk Si, depending on the surface orientations and channel directions. This is because the surface c onfinement induces additional subband splitting and consequently alters the carrier population and the scattering rate. Based on the knowledge of planar MOSFETs, the behavior of strained TG FinFET can be understood and predicted The p iezoresistive property of the GaN HEMT wa s simulated by considering the strain altered 2DEG sheet carrier density and elect ron mobility. It wa s found that the externally applied mechanical stress has negligible effect on the electron density, due to the cancellation of the stress induced piezoelectric polarization in both the AlGaN and GaN layers. Strain is incorporated into an sp 3 d 5 tight binding model to calculate the mobility change under uniaxial and biaxial stress. The simulation result suggests negligible mobility change due to the fact that the GaN conduction band barely warps. T herefore a small gauge factor is expected for GaN HEMT devices By comparing with PAGE 108 108 the experiment al results obtained from a technique eliminating the trapping effect, the best fit set of material parameters is determined. The i mpact of mechanical stress on the gate leakage current in r everse biased GaN HEMT s wa s also investigated A gate leakage model wa s proposed, in which the forward current is due to the Poole Frenkel Emission of electrons from surface states and the backward current is due to the electron diffusion from the AlGaN /GaN interface back to the AlGaN surface states. The simulation and experimental results achieve d close agreement at various temperatures, by fitting several empirical parameters ( E T r, E T and r ) The gate leakage increases (decrease s ) with tensile (compressive) stress and its stress sensitivity decreases at larger reverse bias and higher temperature These results may be explained by the shifting of defect energy level and the altered compensation effect. Finally, the DFT calculation wa s performed to obtain the bulk GaN band structure with correct bandgap. The HSE function al with a 31% mixing parameter wa s determined as the DFT strategy for GaN defect calculation. 7 .2 R ecommendations for Future W ork The GaN HEMT reliability issues are t ightly related to the defects inside the devices. During the GaN HEMT fabrication process, various defects can be introduced into the device. Performing DFT defect calculation on candidate defects can provide useful information such as defect formation e nergy and defect level s By c ombining the gate leakage model ing and the DFT defect calculation, and comparing the simulation results to experiment, the dominant defect s and their origins can be better understood. PAGE 109 109 The GaN HEMTs are conventionally operated under high drain bias. There is a l ar ge piezoelectric stress present in the AlGaN barrier and a large current flows in the channel Investigation of the impacts of mechanical stress in the high power regime can offer better understanding of GaN HEMT degradation mechanisms. PAGE 110 110 A PPENDIX A INCAR FILES There are many setting flags that the VASP users can specify based on their own project needs. In this appendix, the INCAR files for the current work using the standard DFT, LDA+U, and HSE methods are list from Tabl e A 1 to Table A 4 A more detail description of these flags can be found in the VASP u ser manual. Table A 1. The INCAR file for the self consistent (SC) calculation step in the standard DFT model Self Consistent Calculation Step # Staring Parameter for this run ISTART = 0 ICHARG = 2 LWAVE = .TRUE. LCHARG = .TRUE. Start a new job without a WAVECAR as input file. C reate the initial charge density by taking superposition of atomic charge densities. C reate an output WAVECAR file. C reate an output CHGCAR file. # Electronic relaxation PREC = Accurate ENCUT = 500eV NELM = 150 NELMIN = 6 NELMDL = 7 EDIFF = 1E 07 ALGO = Normal LREAL = .FALSE. T his option affects mesh, ENMAX, and the generation of the pseudopotential. Plane wave kinetic energy cutoff. T he maximum number of electronic SC steps. The minimum number of electronic SC steps. S et 7 non self consistent steps at the beginning to improve convergence. S pecifies the stopping criterion for the electronic self consistent loop. S pecifies the algorithm for electronic minimization P rojection is done in the reciprocal space. # Ionic relaxation NSW = 100 IBRION = 2 ISIF = 3 ISYM = 2 EDIFFG = 0.001 ISMEAR = 5 or 0 SIGMA = 0.02 Specifies the maximum number of ionic steps. D etermines how ion positions are updated. Here used the conjugate gradient algorithm. Calculate stress tensor, force. And relax ions, change cell shape and volume. S ymmetry option. S topping criterion for the ionic steps. S pec ifies the partial occupancies for each wavefunction. Set as 5 for unit cell calculation, and 0 for supercell calculation. W idth of the smearing in eV. PAGE 111 111 For the band structure calculation step, only the modified settings are shown in Table A 2. Other flags remain the same as in Table A 1. To incorporate the +U correction in the standard DFT model, the setting list in Table A 3 needs to be added to both Table A 1 and A 2. T o execute the HSE calculation, the hybrid functional settings in Table A 4 need s to be added to both Table A 1 and A 2 Table A 4 also specifies the list of flags that need to be modified. Table A 2. L ist of the modified settings in the INCAR file for the band structure calculation step in the standard DFT model Band Structure Calculation Step ISTART = 1 ICHARG = 11 NSW = 0 IBRION = 1 ISYM = 0 S tart the job using the pre exist WAVECAR file. Keep the charge density constant to obtain eigen values. No ionic relaxation. N o ionic relaxation. N o symmetry. Table A 3. L ist of INCAR flags that defines the +U correction in this work. LDA+U Method LMAXMIC = 4 LDAU = .TRUE. LDAUTYPE = 2 LDAUU = 3.8 LDAUJ = 0.54 E ffective on the d orbital S witches on the LDA+U method. S pecifies the type of LDA+U method. E ffective on site Coulomb interaction parameter. E ffective on site Exchange interaction parameter. Table A 4. L ist of INCAR flags that defines the HSE calculation in this work. HSE Method LHFCALC = .TRUE. HFSCREEN = 0.2 AEXX = 0.31 AGGAX = 0.69 ALDAC = 1 AGGAC = 1 TIME = 0.4 NKRED = 2 S witches on the HF type calculations. S pecifies where to truncate the long range Fock potential. F raction of exact Exchange potential F raction of gradient correction to Exchange potential. F raction of LDA Correlation potential F raction of gradient correctio n of Correlation potential. T he time step. T he k space integral grid reduction factor. # Modified settings in Table A 1 ISTART = 1 EDIFF = 1E 4 IBRION = 1 ISYM = 3 S tart the job using the pre exist WAVECAR file. (also modify this flag in Table A 2) U ses a simple charge mixer Only the stress tensor and force are made symmetry. PAGE 112 112 APPENDIX B POSCAR FILES Figure B 1 shows the POSCAR file for the 4 atom GaN unit cell structure. It contains 2 Ga atoms and 2 N atoms. T he T specifies which basis coordinate of the ions are allowed to move. Figure B 1. The POSCAR file describing a 4 atom GaN unit cell. Figure B 2 shows the POSCAR file for the 32 atom GaN supercell structure. It was constructed from a pre relaxed 4 atom unit cell. In this way, a faster convergence may be achieved. This POSCAR file contains 16 GaN atoms and 16 N atoms. PAGE 113 113 Figure B 2. The POSCAR file describing a 32 atom GaN supercell. PAGE 114 114 APPENDIX C KPOINTS FILES Figure C 1A and B show the k mesh setups for the SC calculation of the unit cell and supercell structures, respectively. A B Figure C 1. The KPOINTS files used in the SC calculation step A) A 4 atom unit cell, and B) a 32 atom supercell. Figure C 2 shows the KPOINTS file that specified a calculation along the M direction. T he band structure along other high symmetric directions can be obtained by modifying the starting and ending nodes. F igure C 2. 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PAGE 130 130 BIOGRAPHICAL SKETCH Min Chu was born in Guangdong Province of China in October 198 2 In 2005, s h e received her Bachelor of Science in electrical e ngineering f rom University of Science and Technology of China Sh e received her Master of Science in 2007 and Doctor of Philosophy in 2011 in the C omputer and E lectrical E ngineering department at the University of Florida H er Ph.D. research focus ed on the impact of strain on piezoresistive properties and reliability of novel device materials and structures 