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13. Band Theory of Solids

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Session Overview

Modules Electronic Materials
Concepts properties of metals and insulators, band theory of solids (Drude; Bloch; Heitler and London), band gaps in metals, semiconductors, and insulators
Keywords metallic bonding, free electron gas, band gap, electrical conductivity, Bloch wave, photoexcitation, charge carrier, metal, insulator, semiconductor, thermal conductivity, valence band, conduction band, antibonding orbital, bonding orbital, carrier mobility, absorption edge, thermal excitation, electron, hole, current, Paul Drude, Felix Bloch, Walter Heitler, Fritz London
Chemical Substances copper (Cu), beryllium (Be), diamond (C), silicon (Si), germanium (Ge), tin (Sn), lead (Pb)
Applications photovoltaics, photosensors, light-emitting diodes (LEDs), temperature sensors

Prerequisites

Before starting this session, you should be familiar with prior topics from Structure of the Atom (Session 1 through Session 7) and Bonding & Molecules (Session 8 through Session 12), specifically:

  • Electron orbital filling order, energy levels, and the Schrödinger equation
  • Linear combination and hybridization of orbitals
  • Electromagnetic radiation, particularly the visible spectrum, and how to convert between wavelength, frequency, and energy

Learning Objectives

After completing this session, you should be able to:

  • Describe the "free electron gas" model and its shortcomings in explaining the physical properties of metals.
  • Derive the band structure of a solid, starting from the orbital diagrams of individual atoms.
  • Calculate the absorption edge, carrier density, and electrical conductivity of a material, and predict how incident photons of given energies or wavelengths will interact with a material.
  • Explain how electronic structure and bonding affects the thermal conductivity, electrical conductivity, optical behavior, and other bulk properties of solids.
  • Classify materials as metals, insulators, or semiconductors, and sketch a schematic band diagram for each one, with key features labeled.
  • Describe what happens during photoexcitation and thermal excitation.

Reading

Archived Lecture Notes #3 (PDF)

Book Chapters Topics
[A&E] 12.5, "Correlation Between Bonding and the Properties of Solids." Ionic solids; molecular solids; covalent solids; metallic solids
[A&E] 12.6, "Bonding in Metals and Semiconductors." Band theory; requirements for metallic behavior; insulators; semiconductors
Amazon logo [JS] 2.4, "The Metallic Bond." Metallic bonding; delocalized electrons; electronegativity
Amazon logo [JS] 15.1, "Charge Carriers and Conduction." Holes and electrons; Ohm's Law; resistivity/conductivity; carrier mobility and drift velocity
Amazon logo [JS] 15.2, "Energy Levels and Energy Bands." Pauli exclusion, Hund's rule, and orbital filling; energy bands and gaps; the Fermi function; thermal promotion; metals, insulators, and semiconductors

Lecture Video

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Resources

Lecture Slides (PDF)

Transcript (PDF)

Lecture Summary

Prof. Ron Ballinger (homepage) gives today's lecture, explaining how the behavior of electrons in aggregate solids determines their electrical and thermal conductivities, optical absorption, and other physical properties. He derives the valence and conduction band structures for electrons in metals (e.g. Cu, Be) using LCAO-MO, and then extends this approach to insulators (e.g. C) and semiconductors (e.g. Si, Ge), which exhibit band gaps. Electrons are promoted across the band gap by photoexcitation or thermal excitation, leaving holes behind. Controlling the population and flow of charge carriers is the fundamental principle underlying modern semiconductor engineering.

Homework

Problems (PDF)

Solutions (PDF)

For Further Study

Textbook Study Materials

See the [A&E] companion website from Pearson for PowerPoint outlines of each chapter, plus online quizzes, interactive graphs and 3D molecular animations:

Supplemental Readings

Bloch, Felix. "Über die Quantenmechanik der Elektronen in Kristallgittern." Zeitschrift für Physik A: Hadrons and Nuclei 52 (1928): 555-600. (Note: this article is in German.)

Heitler, Walter, and Fritz London. "Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik." Zeitschrift für Physik A: Hadrons and Nuclei 44 (1927): 455-472. (Note: this article is in German.)

People

Felix Bloch1952 Nobel Prize in Physics

Paul Drude

Walter Heitler

Fritz London

Other OCW and OER Content

Content Provider Level Notes
Introduction to Semiconductors DoITPoMS Undergraduate See "Introduction to Energy Bands."
5.112 Principles of Chemical Science MIT OpenCourseWare Undergraduate (first-year) Start - 8:15 in Lecture 34: Bonding in Metals and Semiconductors

 

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