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Theory and Modeling of Field Electron Emission from Low-Dimensional

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Theory and Modeling of Field Electron Emission from Low-Dimensional Theory and Modeling of Field Electron Emission from Low-Dimensional Electron Systems by Alex Andrew Patterson B.S., Electrical Engineering University of Pittsburgh, 2011 M.S., Electrical Engineering Massachusetts Institute of Technology, 2013 Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2018 c Massachusetts Institute of Technology 2018. All rights reserved. Author.............................................................. Department of Electrical Engineering and Computer Science January 31, 2018 Certified by. Akintunde I. Akinwande Professor of Electrical Engineering and Computer Science Thesis Supervisor Accepted by . Leslie A. Kolodziejski Professor of Electrical Engineering and Computer Science Chair, Department Committee on Graduate Theses 2 Theory and Modeling of Field Electron Emission from Low-Dimensional Electron Systems by Alex Andrew Patterson Submitted to the Department of Electrical Engineering and Computer Science on January 31, 2018, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering Abstract While experimentalists have succeeded in fabricating nanoscale field electron emit- ters in a variety of geometries and materials for use as electron sources in vacuum nanoelectronic devices, theory and modeling of field electron emission have not kept pace. Treatments of field emission which address individual deviations of real emitter properties from conventional Fowler-Nordheim (FN) theory, such as emission from semiconductors, highly-curved surfaces, or low-dimensional systems, have been de- veloped, but none have sought to treat these properties coherently within a single framework. As a result, the work in this thesis develops a multidimensional, semi- classical framework for field emission, from which models for field emitters of any dimensionality, geometry, and material can be derived. The effects of quantum confinement and emitter tip geometry on the properties of emission were investigated by utilizing the framework to derive models for: i) a highly-curved, nanoscale, metal emitter tip; ii) a bulk silicon emitter with a surface quantum well formed due to electric field penetration and a mechanism that limits the maximum conduction band emitted current density (ECD) to the bulk flux density supply; and iii) a cylindrical silicon nanowire emitter. Results from a highly-curved, nanoscale, metal emitter tip reveal that despite significant electron supply reductions as a result of quantum confinement, the emitted current density (ECD) increases as the emitter radius decreases due to the effects of electric field enhancement. Addition- ally, emitters with radii smaller than 5 nm exhibit a narrow total energy distribution and highly non-linear FN plots. Consistent with experimental observations, the sat- uration of the conduction band ECD in silicon emitters leads to the appearance of three distinct regions in FN plots, which signify conduction-band-dominated, valence- band-dominated, and transitional regimes of emission. Confinement of electrons to a nanowire emitter geometry further reduces the electron supply available for emission and, consequently, the conduction band saturation ECD. Overall, findings show that the dimensionality, geometry, and material of field emitters all play a critical role in field emission processes at the nanoscale. Ac- cordingly, the semiclassical framework for field emission is intended to form a solid foundation upon which more complete models of emission can be developed. Thesis Supervisor: Akintunde I. Akinwande Title: Professor of Electrical Engineering and Computer Science 3 4 Acknowledgments There are many people who have supported me over the course of my time at MIT and I would like to thank them here. For those of you I may not have mentioned, thanks go out to you as well. Foremost, I would like to thank my research advisor and mentor, Professor Tayo Akinwande, for his guidance, encouragement, and constant support. I am especially grateful for the independence he granted me during my time at MIT, particularly in choosing the direction of my research and by allowing me to leave Cambridge to undertake an internship in Germany near the end of my doctoral studies. He has instilled in me the importance of crafting a compelling narrative from my work, the resolve to leave no stone unturned, and the discipline to always \do my due diligence". Under his supervision, I have grown immensely as a researcher and I look forward to his continued mentorship in my career. I would also like to thank my other committee members, Professor Terry Orlando and Professor Dirk Englund, who have provided me with useful feedback during the course of my thesis work. I have known Professor Orlando as long as I have been at MIT; he has guided me as my housemaster, academic counselor, and supervising professor during my teaching assistantship and has supported me in both my research and teaching aspirations. Professor Englund has always advised me to tie the find- ings of my work to experimental results, from which I have developed a valuable understanding of the \other side" of field emission research. From enlightening evenings at the library to the annual IVNC trips, the members of the Akinwande group have been great people to spend my time with daily. In particular, I would like to thank Melissa, Steve, Arash, Neptune, Andy, and Nedo. Thank you as well to others in the MTL community, particularly Dennis and Sabino, for the moments of comic relief and hilarious cookie social discussions. I would also like to thank my friends in Cambridge, and elsewhere, for all of 5 the amazing memories over the years, especially Parker, Joel, Cait, Guha, Joey, Michael, Mike, Dave, Stephen, all the young lads, assorted degases, and the three- time-champion GAME softball team. Finally, I would like to thank my parents, Pat and Steve, my brother Ryan, cousin Dan, cousin Nickie, Aunt Pam, Uncle John, Pete, Ksenia, Sonny, and the rest of my family for their continual support and for always reminding me of where to find home. May we all gather around the shagbark for years to come. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1122374. 6 Contents 1 Introduction 35 2 Background 39 2.1 Current State of Field Emission Theory . 39 2.2 Fowler-Nordheim Theory . 40 2.2.1 Elementary Fowler-Nordheim Equation . 41 2.2.2 Fowler-Nordheim Plots . 44 2.2.3 Standard Fowler-Nordheim Equation: Murphy-Good Theory . 45 2.2.4 Fowler-Nordheim-Type Equations . 47 2.3 Emission from Semiconductors . 48 2.3.1 Emission from the Conduction Band: EF > EC (zs)...... 51 2.3.2 Emission from the Conduction Band: EF < EC (zs)...... 53 2.3.3 Emission from the Valence Band: EF > EV (zs)........ 53 2.3.4 Emission from Surface States . 55 2.3.5 Emission from Accumulation Layer Bound States . 57 2.4 Emission from Highly-Curved Emitter Tips . 58 2.4.1 Field Enhancement Factors . 59 2.4.2 Generalized Fowler-Nordheim Equation for Non-Planar Emitters 60 2.4.3 Semiclassical, Multidimensional Tunneling . 62 2.5 Emission from Low-Dimensional Emitters . 69 7 2.6 Chapter Summary . 73 3 Semiclassical Framework for Field Electron Emission 75 3.1 Introduction . 75 3.2 Semiclassical Solution to the Multidimensional Schr¨odingerEquation 77 3.2.1 Zeroth Order Wave Function: Classical Hamilton-Jacobi Equa- tion . 80 3.2.2 First Order Wave Function: Continuity Equation . 81 3.3 Emitted Current Density Equation . 83 3.3.1 Incident Current Density . 85 3.3.2 Electron Transmission Probability . 86 3.3.3 Bohr-Sommerfeld Quantization . 91 3.3.4 Complete Expression for Emitted Current Density . 93 3.4 Model Assumptions and their Consequences . 93 3.4.1 Quantum Potential . 94 3.4.2 Local Density of States . 96 3.4.3 Semiclassical, Multidimensional Tunneling Model . 98 3.5 Chapter Summary . 100 4 Field Electron Emission from a Low-Dimensional, Highly-Curved Metal Tip 101 4.1 Introduction . 101 4.2 Definition of Paraboloidal Emitter System . 102 4.2.1 Three-Dimensional Parabolic Coordinates . 102 4.2.2 Physical Definition of the Emitter Cathode . 104 4.2.3 Emitter System Electrostatics . 106 4.3 Derivation of Emitted Current Density Equation . 110 4.3.1 Semiclassical Wave Functions in Parabolic Coordinates . 111 8 4.3.2 Bohr-Sommerfeld Bound State Energies: EQ .......... 115 4.3.3 Incident Probability Flux Density: [jQ · n^]........... 116 4.3.4 Electron Transmission Probability: DQ ............. 117 4.3.5 Total Energy Distribution and Total Emitted Current Density 120 4.4 Results and Analysis . 120 4.4.1 Bohr-Sommerfeld Energies . 121 4.4.2 Electronic Subband Origin of Emitted Current Density . 127 4.4.3 Spatial Origin of Emitted Current Density . 132 4.4.4 Emitted Current Density vs. Emitter Radius . 136 4.4.5 Total Energy Distribution . 139 4.4.6 Fowler-Nordheim Plots . 143 4.5 Comparison to Experimental Data: Molybdenum Field Emitter Arrays 147 4.6 Chapter Summary . 151 5 Field Electron Emission from Bulk Silicon 153 5.1 Introduction . 153 5.2 Bulk Silicon Emitter Model . 155 5.2.1 Additional Physical Phenomena in Silicon Field Emitters . 155 5.2.2 Energetic Regimes of Emission from Bulk Silicon . 167 5.2.3 Emitted Current Density Equations . 168 5.2.4 Finite Electronic Flux Supply from Conduction Band States . 172 5.2.5 Approximations and Assumptions . 175 5.3 Results and Analysis . 176 5.3.1 Band Edge Profiles and Bound State Energies . 178 5.3.2 Emitted Current Density vs. Applied Field . 182 5.3.3 Total Energy Distribution . 184 5.3.4 Fowler-Nordheim Plots . 188 5.4 Limitations of the Model . 192 9 5.4.1 Omission of Surface States . 192 5.4.2 Zero Emitted Current Approximation . 194 5.4.3 Zero Internal Current Approximation .
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