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Phys 460 Lecture 14

( pdf version - 6 slides/page )
Monday, October 16, 2006
Lecturer: Richard Martin 		Homework 6

Reading:
Kittel, Chapt. 7

Energy Bands for Electrons in Crystals
Outline

  1. From previous lectures:
    • Part I: Crystal Structures, Diffraction, Reciprocal Lattice, Crystal binding
      Phonons, Dispersion curves,Thermal properties
    • Free electron gas
  2. Recall from last two lectures on the electron gas
    • Chose non-interacting electron gas as the simplest model for electrons
      "Electrons in a box of size L x L x L "
    • Solved Schrodinger Equation for electrons
    • Result: Wavefunction psik(r) = exp (i k r); energy = Ek = (hbar2/2m) k2
    • k determined by boundary condition that electrons fit in the box
      Periodic boundary condition is simplest: leads to kx = integer (2 pi /L), etc.
    • Pauli exclusion principle leads to states being filled up to an energy called the Fermi energy
    • The electron gas is always a metal
  3. The next steps
    • How do we understand that some materials are insulators and some are metals?
    • Are there simple principles that help us understand and even predict which materials will be metals vs which will be insulators?
    • What is a semiconductor? (Answered later.)
  4. The first step: Qualitative ideas that show the effect of the lattice on electrons in one dimension
    • Consider the lattice as a weak effect upon the free electron gas that we studied before
    • Bragg reflection at the zone boundary
    • Leads to energy bands with an energy gap
    • Interpretation of why states at the zone boundary become standing waves with a splitting of the energy to form a gap
    • [Note similarity to phonons!]
  5. Qualitative Picture of he effect of the lattice
    • "Allowed" energy bands - energies of allowed quantum states
    • "Forbidden" energy gaps - energies where there are no allowed quantum states
    • Simple principles that help us understand that a material can be a metal or an insulators
    • More complete discussion next time - more about semiconductors later
  6. Next time
    • Bloch Theorem
    • How bands give us the basis understanding of metals vs. insulators
Email clarification questions and corrections to rmartin@uiuc.edu
Email questions on solving problems to xin2@.uiuc.edu