## ECE 656: Electrothermal Transport in Semiconductors - Fall 2015

### Overview

Fall 2015: Tues, Thurs, 9-10:15AM, KRAN G005

Instructor: M. S. Lundstrom (lundstro at purdue.edu)

TA: Xufeng Wang (wang159 at purdue.edu)

Office Hours: Tuesdays and Thursdays, 8:30-10:00AM, EE-343B

(or make an appointment for a different time by e-mail)

## Course Announcements

Welcome to ECE 656 Fall 2015, Purdue University. All course materials and announcements will be available here. **Check this page frequently**.

**12/14:** Information on the total score and class performance to date is available here.

**12/3:** nanoHUB-U **Nanotechnology: Fundamentals of Nanotransistors** by Prof. Mark Lundstrom starts Jan. 25, 2016. Check here for details.

**12/3:** **Final projects should be e-mailed to Prof. Lundstrom on or before 5:30 PM, Thursday, Dec. 17.**

**12/3: NO CLASS the week of Dec. 8th. Students should wrap up ECE-656 by viewing the on-line lectures for Week 15**

**11/25: Exam 5 grades posted on Blackboard. Exam paper can be picked up from Xufeng @ WANG 2083**

## Course Description

This course is about the flow of charge and heat in semiconductors with an emphasis on transport in nanoscale devices. The objective is to develop a broad understanding of basic concepts. This course is designed for those who work on electronic materials and devices – experimentalists, device physicists, theorists, and computational experts. It is accessible to students with a general, introductory background in semiconductors and addresses the fundamentals of electrothermal transport that every materials and device researcher should be familiar with.

The course consists of four parts.

**Part 1** reviews advanced semiconductor fundamentals concepts covered in 453/595 and 606 and addresses new topics such as phonons and carrier scattering.

**Part 2** examines near-equilibrium transport of electrons and phonons in the presence of small gradients in the electrochemical potential and/ or temperature.

**Part 3** focuses on the Boltzmann Transport Equation (BTE). Some of the results obtained in part 2 are derived from the BTE and the effects of a magnetic field are discussed. The use of Fick’s law of diffusion and Fourier’s Law of heat transport at the nanoscale are discussed.

**Part 4** treats high-field transport in bulk semiconductors, so-called “non-local” transport in nanoscale devices, and provides a gentle introduction to the Non-equilibrium Green’s Function (NEGF) approach to dissipative quantum transport.

Course Texts:

*Near-equilibrium Transport: Fundamentals and Applications* **(NET)** Mark Lundstrom and Changwook Jeong, World Scientific (2013).

(the draft copy of this text is distributed to 656 students and can be downloaded from Blackboard)

Near-Equilibrium Transport Errata

*Fundamentals of Carrier Transport,* 2nd Edition **(FCT)** Mark Lundstrom, Cambridge University Press (2000) ISBN-13: 9780521637244 (paperback)

Fundamentals of Carrier Transport Errata (460 KB)

Fundamentals of Carrier Transport HW Solutions (14 MB)

About this Course:

Course Information (including grading policies) (96 KB)

Course Outcomes (41 KB)

How to take this course (65 KB)

Information on Emergency Preparedness (65 KB)

General References for the Course (67 KB)

For a review of basic semiconductor physics and devices, see the ECE 606 web page

## Part 1: Advanced Semiconductor Fundamentals

### Week 1: Bandstructure Basics (8.25 and 8.27.15)

Reading Assignment: FCT 1.1-1.6, 1.8 / NET 9.1-9.2

Topics: Energy bands and dispersion, DOS and Sums in k-space and energy-space, heterostructures, semiclassical electron dynamics

Week 1 Quiz (280 KB)

Week 1 Quiz Answers (280 KB)

Week 1 HW Assignment (416 KB)

Week 1 HW Solutions

MATLAB Assignment #1. Due at noon on Sep. 4th, 2015. (Submit on blackboard). Good solution chosen by instructor

Week 1: In Class Lectures

Lecture 1: Bandstructure Review

Lecture 2: Semiclassical Transport

Lecture 3: Heterostructure Fundamentals

Lecture 4: Quantum Confinement

Lecture 5: Density of States

Lecture 6: Sums and Integrals

Week 1: Supplementary Online Lectures

Online lecture: ECE 656 Lecture 1: Bandstructure Review

Online lecture: ECE 656 Lecture 33: Heterostructures (Slides 2-11)

Online lecture:ECE 656 Lecture 3: Density of States

Online lecture: ECE 656 Lecture 2: Sums in k-Space/Integrals in Energy Space

Week 1: References and Supplementary Information

Week 1 References (83 KB)

Notes on FD Integrals (214 KB)

Heterostructure Fundamentals (3 MB)

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### Week 2: Introduction to Scattering (9.1 and 9.3.15)

Reading Assignment: FCT 1.7, 2.1-2.3 / NET 6.2

Topics: Fermi’s Golden Rule, scattering rate, characteristic times

Week 2 Quiz (280 KB)

Week 2 Quiz Answers (280 KB)

Week 2 HW Assignment (416 KB)

Week 2 HW Solutions

Week 2: In Class Lectures

Lecture 7: Primer on Carrier Scattering

Week 2: Supplementary Online Lectures

Online lecture: ECE 656 Lecture 21: Scattering and Fermi’s Golden Rule

Online lecture: ECE 656 Lecture 12: Scattering and Transmission (slides 3-11)

Week 2: References and Supplementary Information

For an alternative derivation of FGR, see J.H. Davies, *The Physics of Low-Dimensional Systems*, Cambridge Univ. Press, 1998. Chapter 8, Secs. 8.1 and 8.3.

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**EXAM 1** Tuesday, Sept. 8

Exam 1 Goals (38 KB)

Key Equations for Exam 1 (348 KB)

Exam 1 (ave = 26) (771 KB)

Exam 1 Solutions (866 KB)

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### Week 3: Ionized Impurity Scattering (9.10.15)

Reading Assignment: FCT 2.4

Topics**: **Conwell-Weiskopff and Brooks-Herring approaches

Week 3 Quiz (280 KB)

Week 3 Quiz Answers (280 KB)

Week 3 HW Assignment (416 KB)

Week 3 HW Solutions

Week 3: In Class Lectures

Lecture 8: Fermi's Golden Rule

Lecture 9: Ionized Impurity Scattering

Week 3: Supplementary Online Lectures

Online lecture: ECE 656 Lecture 22: Ionized Impurity Scattering I

Online lecture: ECE 656 Lecture 23: Ionized Impurity Scattering II

Week 3: References and Supplementary Information

For a discussion of modulation doping, screening in 2D, and remote impurity scattering in 2D, see:

J.H. Davies, *The Physics of Low-Dimensional Semiconductors,* Chapter 8, Cambridge Univ. Press, 1998.

For more on overlap integrals, see:

B.K. Ridley, *Quantum Processes in Semiconductors*, 4th Ed., pp. 82-86, Cambridge, 1997.

B.K. Ridley, *Electrons and Phonons in Semiconductor Multilayers*, pp. 60-63, Cambridge, 1997.

D.K. Ferry, *Semiconductors*, pp. 214, 461-464, Macmillan, 1991

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### Week 4: Electron Scattering in Common Semiconductors (Week of 9.15 and 9.17.15)

Reading Assignment: FCT 2.5-2.14

Topics**: **Intervalley and intersubband scattering, scattering in Si and GaAs

Week 4 Quiz (280 KB)

Week 4 Quiz Answers

Week 4 HW Assignment (416 KB)

Week 4 HW Solutions

Week 4: Supplementary Online Lectures

Online lecture: ECE 656 Lecture 23: Phonon Scattering I

Online lecture: ECE 656 Lecture 25: Phonon Scattering II

Online lecture: ECE 656 Lecture 26: Phonon Scattering III

Online lecture: ECE 656 Lecture 27: Scattering in 1D, 2D and 3D

Online lecture: ECE 656 Lecture 27: Scattering of Bloch Electrons

Week 4: References and Supplementary Information

Notes on Scattering and Mobility in 1D, 2D, and 3D

Low Bias Transport in Graphene: An Introduction (3 MB)

### Week 5: Scattering of Phonons (9.22.15)

Reading Assignment: NET 9.6

Topics**: **Normal, Umklapp, and defect scattering

Week 5: In Class Lectures

Lecture 10: Scattering in a Nutshell

Week 5: References and Supplementary Information

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**EXAM 2** Thursday, Sept. 24

Exam 2 Goals (38 KB)

Key Equations for Exam 2 (348 KB)

Exam 2 (771 KB)

Exam 2 Solutions (866 KB)

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## Part 2: Near-Equilibrium Transport

### Week 6: Landauer Approach (9.29 and 10.1.15)

Reading Assignment: NET 2.1-2.7 /FCT 9.9,9.10

Topics**: **Landauer approach, transmission, channels (modes), Fermi-window,and current

Week 6 Quiz (280 KB)

Week 6 Quiz Answers

Week 6 HW Assignment (416 KB)

Week 6 HW Solutions

Week 6: In Class Lectures

Lecture 11: Landauer Approach

Lecture 12: Channels

Lecture 13: Transmission

Lecture 14: Fermi Window and Current

Lecture 15: 1D Resistor

Week 6: Supplementary Online Lectures

Online lecture: ECE 656 Lecture 4: General Model for Transport

Online lecture: ECE 656 Lecture 5: Modes and Transmission

Online lecture: ECE 656 Lecture 12: Scattering and Transmission (Slides 13-39)

Week 6 References and Supplementary Information

For a discussion of “density-of-states”, “conductivity”, and “distribution of modes” effective masses, see these notes: Notes on effective masses (941 KB)

### Week 7: Near-Equilibrium Electron Transport (10.6 and 10.8.15)

Reading Assignment: NET 3.1-3.5

Topics**: **Ballistic to diffusive resistors

Week 7 Quiz (280 KB)

Week 7 Quiz Answers

Week 7 HW Assignment (416 KB)

Week 7 HW Solutions

Week 7: In Class Lectures

Lecture 16: 1D Resistor Discussion

Lecture 17: Landauer Examples

Week 7: Supplementary Online Lectures

Online lecture: ECE 656 Lecture 6: Near-Equilibrium Transport in the Bulk

Online lecture: ECE 656 Lecture 7: Resistance - Ballistic to Diffusive

Online lecture: ECE 656 Lecture 8: More about Resistance

Week 7: References and Supplementary Information

Notes on effective masses (941 KB)

**EXAM 3** Thursday, Oct. 15

Exam 3 Goals (38 KB)

Key Equations for Exam 3 (348 KB)

Exam 3 (771 KB)

Exam 3 Solutions (866 KB)

### Week 8: Near-Equilibrium Phonon Transport (10.20 and 10.22.15)

Reading Assignment: NET 9.3-9.5,9.7

Topics**: **Landauer for phonons transmission, channels (modes), Bose-window, thermal conductivity

Week 8 Quiz

Week 8 Quiz Answers

Week 8 HW Assignment

Week 8 HW Solutions

MATLAB Assignment #2. Due before class on Nov. 12th, 2015. (Submit on blackboard)

Week 8: In Class Lectures

Lecture 18: Phonons

Lecture 19: Heat Transport by Phonons

Week 8: Supplementary Online Lectures

Online lecture: ECE 656 Lecture 13: Phonon Transport

Week 8: References and Supplementary Information

Paper on full band treatment to electron transport http://docs.lib.purdue.edu/nanopub/594/

Paper on full dispersion treatment of phonon transport http://docs.lib.purdue.edu/nanopub/1031/

Paper on thermal transport in bulk and thin-fil Si http://docs.lib.purdue.edu/nanopub/1201/

### Week 9: Thermoelectric Transport (10.27 and 10.29.15)

Reading Assignment: NET 4.1-4.7, 5.1-5.6

Topics**: **Coupled current equations for charge and heat flow by electrons

Week 9 Quiz

Week 9 Quiz Answers

Week 9 HW Assignment

Week 9 HW Solutions

Week 9: In Class Lectures

Lecture 20: Intro to Thermoelectricity

Lecture 21: Charge Current

Lecture 22: Heat Current

Lecture 23: Coupled Current Equations

Lecture 24: TE Devices

Week 9: Supplementary Online Lectures

Online lecture: ECE 656 Lecture 9: Thermoelectric Effects - Charge Flow

Online lecture: ECE 656 Lecture 10: Thermoelectric Effects - (Electronic) Heat Flow

Online lecture: ECE 656 Lecture 11: Coupled Current Equations and Thermoelectric Devices

Week 9: References and Supplementary Information

Paper comparing TE performance in 1D, 2D, and 3D http://docs.lib.purdue.edu/nanopub/539/

### Week 10: Experimental Characterization of Transport (11.3.15)

Reading Assignment: NET 8.1-8.7 /FCT4.7-4.9

Topics**: **Experimental characterization of electronic and heat transport

Week 10 Quiz

Week 10 Quiz Answers

Week 10 HW Assignment

Week 10 HW Solutions

Online lecture: ECE 656 Lecture 17: Near-Equilibrium Measurements I

Online lecture: ECE 656 Lecture 18: Near-Equilibrium Measurements II

Week 10: References and Supplementary Information

**EXAM 4** Thursday, Nov. 5

Exam 4 Goals (38 KB)

Key Equations for Exam 4 (348 KB)

Exam 4 (771 KB)

Exam 4 Solutions (866 KB)

## Part 3: Boltzmann Transport Equation

### Week 11: Boltzmann Transport Equation (BTE) (11.10. and 11.12.15)

Reading Assignment: NET 7.1-7.7 / FCT 3.1-3.5

Topics**: **The BTE, collision integral, near-equilibrium solution for electrons and phonons, magnetic fields, BTE and Landauer

Week 11 Quiz

Weeks 11-12 Quiz Answers

Weeks 11-12 HW Assignment

Weeks 11-12 HW Solutions

Week 11: In Class Lectures

Lecture 25: BTE

Lecture 26: BTE Collision Operator

Lecture 27: Solving the BTE

Online lecture: ECE 656 Lecture 14: The Boltzmann Transport Equation

Online lecture: ECE 656 Lecture 15: The BTE - Transport Coefficients

Online lecture: ECE 656 Lecture 16: The BTE - with B-Field

Online lecture: ECE 656 Lecture 19: Scattering I - Collision Integral

Online lecture: ECE 656 Lecture 20: Scattering II - Relaxation time approximation

Online lecture: ECE 656 Lecture 29: The BTE Revisited - Equilibrium and Ballistic

(slides 1-15 only)

Online lecture: ECE 656 Lecture 17: BTE and Landauer

Week 11: References and Supplementary Information

Week 10 Summary (4 MB)

Notes on Scattering Times and Hall Factors (2 MB)

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### Week 12: The BTE (continued) (11.17 and 11.19.15)

Reading Assignment: Notes and papers will be provided

Topics**: **M-K form of the BTE, relation to diffusion equations, solutions for electron and phonon transport.

Week 12: In Class Lectures

Lecture 28: Solving the BTE II

Lecture 29: Solving the BTE III (B-fields)

Lecture 30: Transmission from BTE

Lecture 31: Solving the Ballistic BTE

Week 12: References and Supplementary Information

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### Week 13: Thanksgiving Week

**EXAM 5** Tuesday, Nov. 24

Exam 5 Goals (38 KB)

Key Equations for Exam 5 (348 KB)

Exam 5 (771 KB)

Exam 5 Solutions (866 KB)

## Part 4: High-field, Non-local, and Quantum Transport

### Week 14: Moments of the BTE and Monte Carlo Simulation (12.1 and 12.3.15)

Reading Assignment: FCT 5.1-5.9 and 6.1-6.9

Topics**: **Balance equations and MC simulation.

Week 14: In Class Lectures

Lecture 32: Moment Equation Approach

Lecture 33: Moment Equations and DD

Lecture 34: TE Moment Equations

Lecture 35: Energy Transport Equations

Lecture 36: Hot Carrier Transport

Lecture 37: Nonlocal Carrier Transport

Online lecture: ECE 656 Lecture 28: Balance Equation Approach I

Online lecture: ECE 656 Lecture 31: Balance Equation Approach II

Online lecture: ECE 656 Lecture 32: Balance Equation Approach III

Online lecture: ECE 656 Lecture 34a: Monte Carlo Simulation I

Online lecture: ECE 656 Lecture 34b: Monte Carlo Simulation II

Week 14: References and Supplementary Information

Week 13 Near equilibrium: Landauer vs. Moments (2 MB)

### Week 15: High-Field, Non-local, and Quantum Transport (12.8 and 12.10.15)

Reading Assignment: FCT 7.1-7.7 and 8.1-8.10

Topics**: **Hot carrier transport in bulk semiconductors, Velocity overshoot, quantum transport.

Online lecture: ECE 656 Lecture 36: High-field Transport

Online lecture: ECE 656 Lecture 33: Non-Local Transport

Online lecture: ECE 656 Lecture 34: Ensemble Effects in Non-Local Transport

Online lecture: ECE 656 Lecture 29: The BTE Revisited - Equilibrium and Ballistic

(slides 16-40 only)

Online lecture: ECE 656 Lecture 39: Ballistic Transport in Devices I

Online lecture: ECE 656 Lecture 35: Introduction to Quantum Transport in Devices

Week 15: References and Supplementary Information

### Finals Week (Week of Dec. 14, 2015)

There will be no final exam.

Projects are due on or before 5:30 PM, Thursday, Dec. 17.