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ECE 874: Physical Electronics

This lecture explores the lattice constant and unit cell of GaAs within the framework of the Kronig-Penney model. It delves into the Block theorem as applied to a 1-D periodic potential and provides example problems involving an electron's movement along the [110] direction. Key topics include the computation of the well width, identification of allowed energy levels, and calculating forbidden energy gaps for the first three energy bands. The presentation offers graphical solutions and emphasizes the significance of periodicity in energy states within crystal systems.

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ECE 874: Physical Electronics

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  1. ECE 874:Physical Electronics Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu

  2. Lecture 14, 16 Oct 12 VM Ayres, ECE874, F12

  3. Three different “a”: - lattice constant - “unit cell” of a periodic potential, p.52 - well width I VM Ayres, ECE874, F12

  4. I Lattice constant a of the Unit cell GaAs: 5.65 Ang “unit cell” a of a 1-D periodic potential Block theorem Well width a Kronig-Penney model for a 1-D periodic potential VM Ayres, ECE874, F12

  5. Example problem: An electron is moving along the [110] direction in GaAs, lattice constant = 5.65 Ang. (a) Write down both versions of the Block theorem explicitly solving for the “unit cell” of the periodic potential in terms of the lattice constant. (b) Draw a model of the transport environment using the Kronig-Penney model where the well width is 20% of the “unit cell” of a periodic potential. Write the dimensions in terms of the lattice constant. +z +x +y VM Ayres, ECE874, F12

  6. Example problem: An electron is moving along the [110] direction in GaAs, lattice constant = 5.65 Ang. [110] Face diagonal distance = ✔2 a Distance between atoms = ✔2 a/2 +z +x +y VM Ayres, ECE874, F12

  7. VM Ayres, ECE874, F12

  8. (b) +z +z +x [110] +x [110] +y +y Rotate [110] to go “straight” VM Ayres, ECE874, F12

  9. b + a = aBl = ✔2 aLC/2 = 3.995 Ang (b) b = 0.8 (aBl = 3.995 Ang) = 3.196 Ang aKP = 0.2 (aBl = 3.995 Ang) = 0.799 Ang b aKP [110] VM Ayres, ECE874, F12

  10. Finite Well boundary conditions, Chp. 02: VM Ayres, ECE874, F12

  11. Finite Well allowed energy levels, Chp. 02: Graphical solution for number and values of energy levels E1, E2,…in eV. a is the finite well width. VM Ayres, ECE874, F12

  12. Similar for Kronig-Penney model but new periodicity requirements: VM Ayres, ECE874, F12

  13. Kronig-Penney model allowed energy levels, Chp. 03: Graphical solution for number and values of energy levels E1, E2,…in eV. a = width of well, b = width of barrier, a + b = Block periodicity aBl VM Ayres, ECE874, F12

  14. Kronig-Penney model allowed energy levels, Chp. 03: Graphical solution for number and values of energy levels E1, E2,…in eV. Also have values for k from RHS. VM Ayres, ECE874, F12

  15. Example problem: (a) What are the allowed (normalized) energies and also the forbidden energy gaps for the 1st-3rd energy bands of the crystal system shown below? (b) What are the corresponding (energy, momentum) values? Take three equally spaced k values from each energy band. VM Ayres, ECE874, F12

  16. k = ± p a + b k = 0 0.5 VM Ayres, ECE874, F12

  17. (a) VM Ayres, ECE874, F12

  18. (b) VM Ayres, ECE874, F12

  19. “Reduced zone” representation of allowed E-k states in a 1-D crystal VM Ayres, ECE874, F12

  20. k = ± p a + b k = 0 VM Ayres, ECE874, F12

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