351-2: Physics of Materials II Bruce Wessels and Peter Girouard Department of Materials Science and Engineering Northwestern University October 1, 2019 Contents 1 Catalog Description (351-1,2)6 2 Course Outcomes6 3 351-2: Solid State Physics6 4 Semiconductor Devices7 4.1 Law of Mass Action..........................7 4.2 Chemistry and Bonding.......................8 4.3 Reciprocal Lattice........................... 11 4.4 Nearly Free Electron Model..................... 11 4.5 Two Level Model........................... 11 5 Band Diagrams 12 5.1 P- and N-Type Semiconductors................... 13 5.2 P-N Junctions............................. 13 5.3 Boltzmann Statistics: Review.................... 13 5.4 P-N Junction Equilibrium...................... 14 5.5 Charge Profile of the p-n Junction.................. 14 5.6 Calculation of the Electric Field................... 15 5.7 Calculation of the Electric Potential................. 16 5.8 Junction Capacitance......................... 17 5.9 Rectification.............................. 18 5.10 Junction Capacitance......................... 19 6 Transistors 19 6.1 pnp Device............................... 20 6.2 pnp Device............................... 21 6.3 Amplifier Gain............................ 21 1 CONTENTS CONTENTS 6.4 Circuit Configurations........................ 22 6.5 MOSFETs................................ 23 6.6 Oxide-Semiconductor Interface................... 23 6.7 Depletion and Inversion....................... 24 6.8 Channel Pinch-Off.......................... 24 6.9 Tunnel Diodes............................. 25 6.10 Gunn Effect.............................. 26 6.11 Gunn Diodes.............................. 26 7 Heterojunctions 26 7.1 Quantum Wells............................ 27 7.2 Heterostructures and Heterojunctions............... 28 7.3 Layered Structures: Quantum Wells................ 28 7.4 Transitions between Minibands................... 29 7.5 Quantum Dots............................. 29 7.6 Quantum Dot Example: Biosensor................. 30 7.7 Quantum Cascade Devices...................... 30 8 Optoelectronics 31 8.1 I-V Characteristics........................... 32 8.2 Power Generation........................... 32 8.2.1 Solar Cell Efficiency..................... 33 8.2.2 Other Types of Solar Cells.................. 33 8.2.3 Metal-Semiconductor Solar Cells.............. 34 8.2.4 Schottky Barrier and Photo Effects............. 34 8.2.5 Dependence of φB on Work Function........... 35 8.2.6 Pinned Surfaces........................ 36 8.2.7 Light Emitting Diodes (LEDs)................ 37 8.2.8 Light Emitting Diodes (LEDs)................ 37 8.2.9 Solid Solution Alloys..................... 37 8.2.10 LED Efficiency........................ 38 9 Lasers 38 9.1 Emission Rate and Laser Intensity................. 39 9.2 Two Level System........................... 39 9.3 Planck Distribution Law....................... 40 9.4 Transition Rates............................ 40 9.5 Two Level System........................... 41 9.6 Three Level System.......................... 41 9.7 Lasing Modes............................. 45 9.8 Laser Examples............................ 45 9.8.1 Ruby.............................. 45 9.8.2 Others Examples....................... 46 9.9 Threshold Current Density...................... 46 9.10 Comparison of Emission Types................... 48 2 CONTENTS CONTENTS 9.11 Cavities and Modes.......................... 48 9.11.1 Longitudinal Laser Modes.................. 49 9.11.2 Transverse Laser Modes................... 49 9.12 Semiconductor Lasers........................ 49 9.13 Photonic Bandgap Materials..................... 50 10 Band Diagrams 50 10.1 Band Diagrams............................ 51 10.2 Heterojunctions and the Anderson Model............. 51 10.3 Band Bending at p-n Junctions................... 51 10.3.1 Calculate the Conduction Band Discontinuity...... 52 10.3.2 Calculate the Valence Band Discontinuity......... 53 10.3.3 Example: p-GaAs/n-Ge................... 54 11 Dielectric Materials 54 11.1 Macroscopic Dielectric Theory.................... 55 11.2 Microscopic Structure......................... 56 11.2.1 Relation of Macroscopic to Microscopic.......... 57 11.3 Polarizability.............................. 57 11.4 Polarization in Solids......................... 57 11.5 Dipole Moment............................ 58 11.6 Polarizability of Solids........................ 59 11.7 Dielectric Constant.......................... 59 11.8 Claussius Mossotti Relation..................... 60 11.9 Frequency Dependence........................ 60 11.10Quantum Theory of Polarizability................. 62 11.11Ferroelectrics.............................. 62 12 Phase Transitions 63 12.1 Lattice Instabilities.......................... 63 12.2 Curie Weiss Law............................ 63 12.3 Ferroelectrics.............................. 64 12.3.1 First versus Second Order Transitions........... 66 12.3.2 Ferroelectric Example: BaTiO3 ............... 66 12.4 Other Instabilities........................... 67 12.5 Piezoelectrics............................. 67 13 Diamagnetism and Paramagnetism 68 13.1 Diamagnetism............................. 69 13.2 Paramagnetism............................ 70 13.2.1 Calculation of Susceptibility................. 72 13.2.2 Calculation of Total Angular Momentum J ........ 73 13.2.3 Spectroscopic Notation.................... 73 13.2.4 Paramagnetic Susceptibility................. 73 13.2.5 Calculation of J ........................ 73 13.2.6 Spin Orbit Interactions.................... 74 3 CONTENTS CONTENTS 13.2.7 Effective Magnetic Number................. 74 13.2.8 Paramagnetic Properties of Metals............. 75 13.2.9 Band Model.......................... 75 13.2.10 Multivalent Effects...................... 76 14 Ferromagnetism 77 14.1 Ferromagnetic Phase Transition................... 77 14.2 Molecular Field............................ 78 14.2.1 Prediction of TC ........................ 79 14.2.2 Temperature Dependence of M(T ) for Ferromagnetism. 80 14.2.3 Ferromagnetic-Paramegnetic Transition.......... 80 14.2.4 Ferromagnetic-Paramagnetic Transition.......... 80 14.2.5 Ferromagnetic-Paramagnetic Transition.......... 81 14.2.6 Ferromagnetic-Paramagnetic Transition.......... 81 14.2.7 Ferromagnetism of Alloys.................. 82 14.2.8 Transition Metals....................... 82 14.2.9 Ni Alloys........................... 82 14.2.10 Band Model.......................... 83 14.2.11 Spin Waves.......................... 83 14.2.12 Magnon Dispersion..................... 84 14.3 Ferrimagnetic Order......................... 84 14.3.1 Magnetic Oxides....................... 85 14.3.2 Magnetization and Hysteresis................ 86 14.4 Domains and Walls.......................... 86 14.4.1 Energy of Bloch Domain Walls............... 87 14.5 Anisotropy of Magnetization.................... 87 14.5.1 Anisotropy of Magnetization................ 88 14.6 Ferrimagnetic Ordering....................... 88 14.6.1 Exchange Terms and Susceptibility............. 89 14.6.2 Structure Dependence of Jex ................ 89 15 Optical Materials 89 15.1 Frequency Doubling......................... 90 15.2 Nonlinearity in Refractive Index.................. 90 15.3 Electro-Optic Modulators...................... 91 15.4 Optical Memory Devices....................... 91 15.5 Volume Holography......................... 92 15.6 Photorefractive Crystals....................... 92 15.7 Photorefractive Crystals....................... 93 15.8 All-Optical Switching......................... 93 15.9 Acousto-Optic Modulators...................... 94 15.10Integrated Optics........................... 94 15.11Dielectric Waveguides........................ 95 15.12Phase Shifter.............................. 95 15.13LiNbO Phase Shifter......................... 96 4 CONTENTS CONTENTS 16 Superconductivity 96 16.1 BCS Theory.............................. 96 16.2 Density of States and Energy Gap.................. 97 16.3 Heat Capacity............................. 97 16.4 Superconductor Junctions...................... 98 16.5 Junctions................................ 98 16.6 Type I and Type II Superconductors................ 98 16.7 High TC Superconductors...................... 99 16.8 Cuprate Superconductors...................... 99 17 351-2 Problems 100 18 351-2 Laboratories 101 18.1 Laboratory 1: Measurement of Charge Carrier Transport Param- eters Using the Hall Effect...................... 101 18.1.1 Objective............................ 101 18.1.2 Outcomes........................... 102 18.1.3 References........................... 102 18.1.4 Pre-Lab Questions...................... 102 18.1.5 Experimental Details..................... 103 18.1.6 Instructions/Methods.................... 104 18.1.7 Link to Google Form for Data Entry............ 104 18.1.8 Lab Report Template..................... 104 18.1.9 Hints for derivation..................... 105 18.2 Laboratory 2: Diodes......................... 106 18.2.1 Objective............................ 106 18.2.2 Outcomes........................... 106 18.2.3 Pre-lab Questions....................... 106 18.2.4 References........................... 107 18.3 Laboratory 3: Transistors....................... 110 18.3.1 Objective............................ 110 18.3.2 Outcomes........................... 110 18.3.3 Pre-lab questions....................... 110 18.3.4 Experimental Details..................... 111 18.4 Laboratory 4: Dielectric Materials.................. 113 18.4.1 Objective............................ 113 18.4.2 Outcomes..........................