Lessons from Nanoscience: A Lecture Note Series

Transcription

1 Lessons from Nanoscience: A Lecture Note Series Volume 1: Lessons from Nanoelectronics: A New Perspective on Transport Supriyo Datta Purdue University datta@purdue.edu For updates and discussion related to these notes, readers are encouraged to check our website at Readers interested in gaining access to online video lectures based on these notes, can check the nanohub-u website at World Scientific Publishing Company i

2 ii Preface Preface Everyone is familiar with the amazing performance of a modern smartphone, powered by a billion-plus nanotransistors, each having an active region that is barely a few hundred atoms long. These lectures, however, are about a less-appreciated by-product of the microelectronics revolution, namely the deeper understanding of current flow, energy exchange and device operation that it has enabled, which forms the basis for what we call the bottom-up approach. I believe these lessons from nanoelectronics should be of broad relevance to the general problems of non-equilibrium statistical mechanics which pervade many different fields. To make these lectures accessible to anyone in any branch of science or engineering, we assume very little background beyond linear algebra and differential equations. We hope to reach all those with an interest in basic physics, even if they are not specializing in devices or transport theory. For dedicated graduate students and the specialists, I have written extensively in the past. But they too may enjoy these notes taking a fresh look at a familiar subject, emphasizing the insights from mesoscopic physics and nanoelectronics that are of general interest and relevance. Finally I should stress that these are lecture notes representing work in progress with typos included to enhance the readers attention, as my colleague Gerhard likes to put it. With your feedback and suggestions, I hope to have a better version in the future that requires less attention! Until then we are using the website to post corrections, revisions, video lectures, tutorials, quizzes and also to host a Q&A forum based on questions from readers. April 21, 2012

3 Contents Preface ii I: The New Ohm s Law 1. The Bottom-Up Approach Why Electrons Flow The Elastic Resistor Ballistic and Diffusive Transport Conductivity Diffusion Equation for Ballistic Transport What about Drift? Electrostatics is Important Smart Contacts 103 II: Old Topics in New Light 10. Thermoelectricity Phonon Transport Measuring Electrochemical Potentials Hall Effect Spin Valve Kubo Formula Second Law Fuel Value of Information 238 III: Contact-ing Schrodinger 18. The Model Non-Equilibrium Green s Functions (NEGF) Can Two Offer Less Resistance than One? Quantum of Conductance Rotating an Electron Does NEGF Include Everything? The Quantum and the Classical 390 References / Further Reading 403 Appendices 411

4 iv Detailed Contents Detailed Contents Preface ii I: The New Ohm s Law 1. The Bottom-Up Approach Why Electrons Flow Two Key Concepts 2.2. Fermi Function 2.3. Non-equilibrium: Two Fermi Functions 2.4. Linear Response 2.5. Difference in Agenda Drives the Flow 3. The Elastic Resistor How an Elastic Resistor Dissipates Heat 3.2. Conductance of an Elastic Resistor 3.3. Why an Elastic Resistor is Relevant 4. Ballistic and Diffusive Transport Ballistic and Diffusive Transfer Times 4.2. Channels for Conduction 5. Conductivity E(p) or E(k) Relations 5.2. Counting States 5.3. Drude Formula 5.4. Is Conductivity proportional to Electron Density? 5.5. Quantized Conductance 6. Diffusion Equation for Ballistic Transport Electrochemical Potentials Out of Equilibrium 6.2. Currents in Terms of Non-Equilibrium Potentials 7. What about Drift? Boltzmann Transport Equation, BTE 7.2. Diffusion Equation from BTE 7.3. Equilibrium Fields Do Matter 7.4. The Two Potentials

5 v Detailed Contents 8. Electrostatics is Important The Nanotransistor 8.2. Why the Current Saturates 8.3. Role of Charging 8.4. Rectifier Based on Electrostatics 8.5. Extended Channel Model 9. Smart Contacts Why p-n Junctions are Different 9.2. Contacts are Fundamental II: Old Topics in New Light 10. Thermoelectricity Seebeck Coefficient Thermoelectric Figures of Merit Heat Current Delta Function Thermoelectric 11. Phonon Transport Phonon Heat Current Thermal Conductivity 12. Measuring Electrochemical Potentials The Landauer Formulas Büttiker Formula 13. Hall Effect Why n- and p- Conductors Are Different Angular Profile of Electrochemical Potential Measuring the Potential Non Reciprocal Circuits 14. Spin Valve Mode Mismatch and Interface Resistance Spin Potentials Spin-Torque Polarizers and Analyzers 15. Kubo Formula Kubo Formula for an Elastic Resistor Onsager Relations

6 vi Detailed Contents 16. Second Law Asymmetry of Absorption and Emission Entropy Law of Equilibrium Fock Space States Alternative Expression for Entropy 17. Fuel Value of Information Information-Driven Battery Fuel Value Comes From Knowledge Landauer s Principle Maxwell s Demon III: Contact-ing Schrodinger 18. The Model Schrödinger Equation Electron-Electron Interactions Differential to Matrix Equation Choosing Matrix Parameters 19. Non-Equilibrium Green s Functions (NEGF) One-level Resistor Multi-level Resistors Conductance Functions for Coherent Transport Elastic Dephasing 20. Can Two Offer Less Resistance than One? Modeling 1D Conductors Quantum Resistors in Series Potential Drop Across Scatterer(s) 21. Quantum of Conductance D Conductor as 1D Conductors in Parallel Contact self-energy for 2D Conductors Quantum Hall Effect 22. Rotating an Electron One-level Spin Valve Rotating Magnetic Contacts Spin Hamiltonians

7 vii Detailed Contents Vectors and Spinors Spin Precession From NEGF to Diffusion 23. Does NEGF Include Everything? Coulomb Blockade Fock Space Description Strong Correlations Entanglement 24. The Quantum and the Classical Spin coherence Pseudo-spins Quantum Entropy Does Interaction Increase the Entropy? Spins and magnets References / Further Reading 403 Appendices 412 A. Fermi and Bose Function Derivatives B. Angular Averaging C. Hamiltonian with E- and B-Fields D. Transmission Line Parameters from BTE Equations E. NEGF Equations F. MATLAB Codes for Text Figures

8 viii Some Symbols Used Some Symbols Used Some Symbols Used Constants Electronic charge - q - 1.6e-19 coul. Unit of Energy 1 ev + 1.6e-19 Joules Planck's constant h = h / 2π 6.626e -34 Joule-sec 1.055e -34 Joule-sec Boltzmann constant k 1.38e-23 Joule / K ~ 25 mev / 300K Free electron mass m e-31 Kg Effective mass m Other Symbols I Electron Current amperes (A) (See Fig.3.2) J Electron Current density A/m 2 V Electron Voltage volts (V) U Electrostatic Potential ev µ Electrochemical Potential ev (also called Fermi level or quasi-fermi level) µ mob Mobility m 2 /V-sec R Resistance Ohms (V/A) G Conductance Siemens (A/V) G(E) Conductance at 0K with µ 0 =E Siemens (A/V) D µ Diffusivity Mobility m 2 /sec m 2 /V-sec

9 ix ρ Some Symbols Used Resistivity Ohm-m (3D), Ohm (2D) σ Conductivity S/m (3D), S (2D) A Area m 2 W Width m L Length m E Energy ev f (E) Fermi Function Dimensionless f E Thermal Broadening Function (TBF) / ev kt f E Normalized TBF Dimensionless D(E) Density of States /ev N(E) Number of States with Energy < E Dimensionless Equals Number of Electrons at 0K with µ 0 =E n Electron Density (3D or 2D or 1D) /m 3 or /m 2 or /m M(E) Number of Channels Dimensionless (also called transverse modes) T Temperature degrees Kelvin (K) t Transfer Time seconds ν Transfer Rate /second γ ν Energy Broadening ev [X] + Complex conjugate of transpose of matrix [X] H (Matrix) Hamiltonian ev G R (E) (Matrix) Retarded Green s function /ev G A (E) = [G R (E)] + (Matrix) Advanced Green s function /ev G n (E) / 2π (Matrix) Electron Density /ev, per gridpoint A(E) / 2π (Matrix) Density of States /ev, per gridpoint Γ(E) (Matrix) Energy Broadening ev

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11 Acknowledgements Thanks to World Scientific Publishing Corporation and, in particular, our series editor, Zvi Ruder for joining us in this partnership. The precursor to this lecture note series, namely the Electronics from the Bottom Up initiative on was funded by the U.S. National Science Foundation (NSF), the Intel Foundation, and Purdue University. The nanohub-u recently offered its first online course based on these notes and I am thankful for the feedback I received from many online students whom I have never met. We gratefully acknowledge Purdue and NSF support for this program, along with the superb team of professionals who made nanohub-u a reality. A special note of thanks to Mark Lundstrom for his leadership that made it all happen and for his encouragement and advice. I am indebted to Ashraf Alam, Kerem Camsari, Deepanjan Datta, Vinh Diep, Samiran Ganguly, Seokmin Hong, Changwook Jeong, Bhaskaran Muralidharan, Angik Sarkar, Srikant Srinivasan for their valuable feedback and suggestions. Finally I would like to express my deep gratitude to all who have helped me learn, a list that includes not only those named above, but also many teachers, colleagues and students over the years, starting with the late Richard Feynman whose classic lectures on physics, I am sure, have inspired many like me. 3

12 4 Lessons from Nanoelectronics To Malika, Manoshi and Anuradha