QUANTUM ELECTRONIC DEVICE SIMULATIONA DISSERTATIONSUBMITTED TO THE DEPARTMENT OF ELECTRICAL ENGINEERINGAND THE COMMITTEE ON GRADUATE STUDIESOF STANFORD UNIVERSITYIN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR THE DEGREE OFDOCTOR OF PHILOSOPHYBryan A. BiegelMarch 1997
Copyright by Bryan A. Biegel 1997All Rights Reservedii
AbstractAn accurate understanding of quantum wave effects in electronic devices is importantfor several reasons. In the short term, this understanding will enable the suppression ofthese increasingly significant parasitic effects in ever-smaller conventional devices. In themedium term, this understanding will enable the control of these effects, possibly extending down-scaling closer to the quantum realm with hybrid conventional-quantum electronic devices. In the longer term, an understanding of quantum electronic effects isnecessary for the possible development of a true quantum device technology, with thepotential for much greater functionality per unit cost, size, and power. To build this understanding, a numerical quantum device simulator called SQUADS (Stanford QUAntumDevice Simulator) was developed. This dissertation describes the implementation, capabilities, and some illustrative simulation results of SQUADS.The design of SQUADS was directed by two goals: the study of quantum device operation, and the study of quantum device simulation. In pursuing these goals, a comprehensive 1-dimensional simulation tool was developed for modeling quantum-effect electronicsystems of arbitrary structure. Two independent formulations of quantum mechanics wereimplemented in SQUADS. The first is the widely-employed transfer-matrix method ofquantum system simulation, which provides a source of quick initial simulation results,and is especially useful in detailing the energy spectrum of carriers in the device. The second method uses the Wigner function formulation of quantum mechanics, which is morecomputationally intensive, but which allows a more intuitive and complete description ofreal quantum electronic systems, especially including transient response and energy dissipation.In addition to describing the basic implementation and simulation results of these sim-iii
ulation methods in SQUADS, this thesis also describes three detailed investigations ofquantum device simulation and operation, using SQUADS as the simulator and the resonant tunneling diode as the test device. An investigation of self-consistency in quantumdevice simulation found that both the efficient steady-state and the more accurate transientself-consistency iteration methods have important roles to play, and that a Gummel (asopposed to full-Newton) iteration method is almost always quite adequate. An investigation of the effect of slew rate variation in transient RTD simulation showed that the use ofan appropriate applied bias slew rate is necessary for accurate simulations and to preventthe misinterpretation of simulation results. Finally, the detailed simulation investigation ofthe physics of an RTD produced a better understanding of this device, corrected severalerrors in previous interpretation of simulation and experimental results, and resulted inimproved agreement between simulation and experiment for this device.In general, this work found that quantum device simulation is still in a formative stage,although significant advances have been made in this work and elsewhere. Quantumdevice simulation is not yet at a point where it can reliably reproduce or predict quantitative experimental results, whether because of non-idealities in experiment or inaccuracy ofthe simulator. Nevertheless, quantum device simulation in this work and elsewhere hasalready contributed to the debates surrounding significant unresolved issues of quantumdevice physics and operation.iv
AcknowledgmentsI am happy to have this opportunity to acknowledge the assistance and support of themany people who contributed to the completion of this work and to my enjoyable andrewarding experience at Stanford. First, I would like to thank my research advisor, Professor James Plummer, who always supported and encouraged my research efforts, in spite ofthe long duration of this project. His broad understanding of the state-of-the-art and futureof electronics made it possible for me to pursue my rather esoteric interests in quantumelectronic devices while still obtaining useful guidance from him. At the same time, heallowed me to develop my own research abilities, make my own contributions, and learnself-sufficiency. I believe all Ph.D. graduates from Stanford should have this experience.I would also like to thank my associate advisor, Professor Walter Harrison, for his generosity of time and candid discussions. Professor Harrison posed very thought-provokingquestions about this work, prompting worthwhile examinations of subtle underlyingassumptions. I appreciate his deep understanding of quantum physics, and his unique ability to make that knowledge accessible to others. Dr. Zhiping Yu has also been very helpfulin sharing his knowledge of semiconductor device physics and simulation. Along withProfessor Plummer, Professor Harrison, and Dr. Yu, I would like to acknowledge Professor Lambertus Hesselink as the final member of my reading committee. They have given along and much-appreciated effort in the review and correction of this dissertation.The time and assistance of many other people in the completion of this work shouldalso be acknowledged. My orals committee; including Professors Plummer, Hesselink,Robert Dutton, and James S. Harris, as well as Dr. Zhiping Yu; were very gracious inaccommodating my schedule and subject, and their insightful questions and commentswere much appreciated. Dr. Kevin Jensen was always willing to discuss my latest discov-v
eries and challenges. He provided key insights that led to some of the significant contributions of this work. Dr. Kiran Gullapalli and Dr. William Frensley also provided valuabletechnical input during this research. Dr. Daniel Scales was my main programming guru;able to help slay the hardiest of software bugs. Many people assisted during the briefexperimental portion of this work, including Chan-Hong Chern and Vincent Arbet atUCLA, who grew SiGe MBE samples for me pro-bono; and all of the Integrated CircuitsLaboratory technicians, who make useful work therein possible.The information systems personnel at the Center for Integrated Systems, includingCharlie Orgish, Laura Schrager, and Dr. Ernest Wood, were always willing and able toaddress computer-related problems and questions. Robert Taft, Walter Snoeys, WilliamWong, and Amit Paul also generously gave their time to make computing resources in thePlummer group as productive as possible. Vital to the success of this project were theimpressive computing resources available to Stanford students - a reflection of Stanford’sforesight in procuring and managing these resources. The Plummer administrative assistants over the years, including Joyce Pelzl, Susan Stout, and Jane Edwards, have been uniformly friendly and knowledgeable about how to get things done.Finally, I would like to mention some of the many people who helped to make my timeat Stanford enjoyable and rewarding. I am happy to call many Plummer students myfriends, including Eric Perozziello, Walter Snoeys, Robert Taft, Mary Weybright, andTiemin Zhou. Along with Scott Gallert, Jim Roche, Dan Scales, and many others, theymade me wish my time at Stanford would never end. The support and encouragement ofmy long-time friend, Margaret O’Hare, has always been appreciated, though not oftenacknowledged. And, of course, I appreciate the love and support of my family: my parentsPeter and JoAnn, and my siblings Denise, David, Mark, Glen, Andrea, Miriam, Alex, andJosh. My parents-in-law, Mel and Fran Holdener, gave me a home away from home, andhave never failed to provide support (nutritional and emotional) on a regular basis (especially at holidays). Last and certainly not least, I want to recognize my wife, Teresa, forher superhuman endurance and love during one of the longest doctoral programs known toman (or woman). I’m sure she sometimes wondered whether I did still recognize her, buthaving her picture in my wallet helped.Support for this research was provided in part by the Joint Services Electronics Program and by the Computational Prototyping Program at Stanford University.vi
ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.1Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.21.1.1 The Quantum Challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.1.2 The Quantum Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.1.3 More Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.1.4 Further Possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Approach and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124561.3Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Overview of Quantum Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1Past, Present, and Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.22.1.1 The Genealogy of Quantum Electronics . . . . . . . . . . . . . . . . . . . . . . .2.1.2 The Optical Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1.3 The Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Phenomena and Structures for Quantum Devices . . . . . . . . . . . . . . . . . . . . .2.32.2.1 Basic Quantum Wave Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.2 Basic Wave Components and Quantum Structures . . . . . . . . . . . . . . 17Devices for Quantum Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20vii12121517
2.42.3.1 General Concepts of Quantum Devices . . . . . . . . . . . . . . . . . . . . . . .2.3.2 Quasi-Equilibrium Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.3 Far-From-Equilibrium Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.4 Prototype Quantum Electronic Device . . . . . . . . . . . . . . . . . . . . . . . .Quantum Electronic and Quantum Computing Systems . . . . . . . . . . . . . . . .20212324252.52.4.1 Architecture Challenges and Conclusions . . . . . . . . . . . . . . . . . . . . .2.4.2 The Cellular Automaton Architecture . . . . . . . . . . . . . . . . . . . . . . . .2.4.3 The Quantum Filter Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.4.4 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2627293233References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343Quantum Device Simulation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1State of Quantum Device Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2Goals of Quantum Device Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3Classical Electronic Device Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.43.3.1 The Boltzmann Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . .3.3.2 Strengths of the BTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3.3 Implications for Quantum Device Simulation . . . . . . . . . . . . . . . . . .Quantum Transport Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4.13.4.243444546Relationships Between Candidate Formulations . . . . . . . . . . . . . . . . 46Analysis of Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4.2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4.2.2The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4.2.3The Transfer-Matrix Method . . . . . . . . . . . . . . . . . . . . . . . . 493.4.2.4The Quantum Transport Equation Approaches . . . . . . . . . . 493.4.2.5Derivatives of the Wigner Function Method . . . . . . . . . . . . 523.5SQUADS Simulation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.63.5.1 One-Dimensional versus Multi-Dimensional . . . . . . . . . . . . . . . . . . .3.5.2 Envelope Function versus Tight-Binding . . . . . . . . . . . . . . . . . . . . . .3.5.3 Two-Tiered Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.5.4 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.5.5 Research Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .viii535355565757
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584The Transfer-Matrix Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.1History and State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2.14.2.24.2.34.2.4General Solutions of the Schrödinger Equation . . . . . . . . . . . . . . . . .Gridded Potential Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Wavefunction Matching Conditions . . . . . . . . . . . . . . . . . . . . . . . . . .System Transmission Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.4.1 Basic STM Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.4.24.3STM Calculation Complications . . . . . . . . . . . . . . . . . . . . . 74Quantum Device Simulation Using the TMM . . . . . . . . . . . . . . . . . . . . . . . . 764.3.14.3.24.46567717273Current-Voltage Curve Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.3.1.1 Determining the Transmission Amplitude . . . . . . . . . . . . . . 764.3.1.2Calculating Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.3.1.3I-V Curve Simulation Overview . . . . . . . . . . . . . . . . . . . . . 80Calculating the Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3.2.1 Basic Wavefunction Calculation . . . . . . . . . . . . . . . . . . . . . 824.3.2.2Classically Forbidden T-Contact . . . . . . . . . . . . . . . . . . . . . 834.3.2.3Quantum Turning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.3.3 Calculating the Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3.4 Calculating the Carrier Density Profile . . . . . . . . . . . . . . . . . . . . . . . .4.3.5 Calculating the Wigner Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .Alternative Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.4.14.4.287889091Node-Centered Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Alternate STM Calculation Algorithms . . . . . . . . . . . . . . . . . . . . . . . 934.4.2.1 Region Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.4.2.2Normalization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.54.4.3 Piece-Wise Linear Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.64.5.1 Accuracy of Linear versus Constant Potential Regions . . . . . . . . . . . 984.5.2 Efficiency of STM Calculation Algorithms . . . . . . . . . . . . . . . . . . . . 1014.5.3 Constant versus Variable Effective Mass . . . . . . . . . . . . . . . . . . . . . . 102Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103ix
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055The Wigner Function Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.1History and State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.2Analytical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.35.2.1 The Wigner Function Transport Equation . . . . . . . . . . . . . . . . . . . . . 1115.2.2 Gridding and the Potential Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.2.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.2.4 Carrier and Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.3.15.3.25.3.3Discretization of the Independent Variables . . . . . . . . . . . . . . . . . . . . 117The Discrete WFTE Matrix Equation . . . . . . . . . . . . . . . . . . . . . . . . . 119Discretization of the WFTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.3.3.1 Short-Hand Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.3.3.2Diffusion Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.3.3.3Drift Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.3.3.4Scattering Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.3.3.5Transient Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.3.3.6The Discrete WFTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.45.3.4 Discrete Carrier and Current Densities . . . . . . . . . . . . . . . . . . . . . . . . 131Efficient Solution of the Discrete WFTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.55.4.1 Memory Management Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.4.2 Computation Time and Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.4.3 Other Solution Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.65.5.1 Gaussian Wave Packet Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 1425.5.2 Diffusion Term Discretization Comparison . . . . . . . . . . . . . . . . . . . . 1445.5.3 Transient Approach Comparison . . . . . . . . . . . . . . .
electronic devices while still obtaining useful guidance from him. At the same time, he allowed me to develop my own research abilities, make my own contributions, and learn self-sufficiency. I believe all Ph.D. graduates from Stanford should have this experience. I would also like to thank my associate advisor, Professor Walter Harrison, for .
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According to the quantum model, an electron can be given a name with the use of quantum numbers. Four types of quantum numbers are used in this; Principle quantum number, n Angular momentum quantum number, I Magnetic quantum number, m l Spin quantum number, m s The principle quantum
Quantum computing is a subfield of quantum information science— including quantum networking, quantum sensing, and quantum simulation—which harnesses the ability to generate and use quantum bits, or qubits. Quantum computers have the potential to solve certain problems much more quickly t
The Quantum Nanoscience Laboratory (QNL) bridges the gap between fundamental quantum physics and the engineering approaches needed to scale quantum devices into quantum machines. The team focuses on the quantum-classical interface and the scale-up of quantum technology. The QNL also applies quantum technology in biomedicine by pioneering new
For example, quantum cryptography is a direct application of quantum uncertainty and both quantum teleportation and quantum computation are direct applications of quantum entanglement, the con-cept underlying quantum nonlocality (Schro dinger, 1935). I will discuss a number of fundamental concepts in quantum physics with direct reference to .
1.3.7 Example: quantum teleportation 26 1.4 Quantum algorithms 28 1.4.1 Classical computations on a quantum computer 29 1.4.2 Quantum parallelism 30 1.4.3 Deutsch's algorithm 32 1.4.4 The Deutsch-Jozsa algorithm 34 1.4.5 Quantum algorithms summarized 36 1.5 Experimental quantum information processing 42 1.5.1 The Stern-Gerlach experiment 43
Quantum effects - superposition, interference, and entanglement NISQ - Noisy Intermediate-Scale Quantum technology, often refers in the context of modern very noisy quantum computers QASM - Quantum Assembly used for programming quantum computers Quantum supremacy - demonstration of that a programmable quantum
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