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ECE 874: Physical Electronics

ECE 874: Physical Electronics. Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu. Lecture 15, 18 Oct 12. Example problem:

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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 15, 18 Oct 12 VM Ayres, ECE874, F12

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

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

  5. (a) VM Ayres, ECE874, F12

  6. (b) VM Ayres, ECE874, F12

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

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

  9. (b) VM Ayres, ECE874, F12

  10. “Reduced zone” representation of allowed E-k states in a 1-D crystal This gave you the same allowed energies paired with the same momentum values, in the opposite momentum vector direction. Always remember that momentum is a vector with magnitude and direction. You can easily have the same magnitude and a different direction. Energy is a scalar: single value. VM Ayres, ECE874, F12

  11. Can also show the same information as an “Extended zone representation” to compare the crystal results with the free carrier results. Assign a “next” k range when you move to a higher energy band. VM Ayres, ECE874, F12

  12. Example problem: There’s a band missing in this picture. Identify it and fill it in in the reduced zone representation and show with arrows where it goes in the extended zone representation. VM Ayres, ECE874, F12

  13. The missing band: Band 2 VM Ayres, ECE874, F12

  14. VM Ayres, ECE874, F12

  15. Notice that upper energy levels are getting closer to the free energy values. Makes sense: the more energy an electron “has” the less it even notices the well and barrier regions of the periodic potential as it transports past them. VM Ayres, ECE874, F12

  16. Note that at 0 and ±p/(a+b) the tangent to each curve is flat: dE/dk = 0 VM Ayres, ECE874, F12

  17. A Brillouin zone is basically the allowed momentum range associated with each allowed energy band Allowed energy levels: if these are closely spaced energy levels they are called “energy bands” Allowed k values are the Brillouin zones Both (E, k) are created by the crystal situation U(x). The allowed energy levels are occupied – or not – by electrons VM Ayres, ECE874, F12

  18. VM Ayres, ECE874, F12

  19. (b) VM Ayres, ECE874, F12

  20. VM Ayres, ECE874, F12

  21. What happens to the e- in response to the application of an external force: example: a Coulomb force F = qE (Pr. 3.5):  VM Ayres, ECE874, F12

  22. (d) VM Ayres, ECE874, F12

  23. (d) Conduction energy bands Symmetric <111> type 8 of these <100> type 6 of these [100] [100] Warning: you will see a lot of literature in which people get careless about <direction type> versus [specific direction] VM Ayres, ECE874, F12

  24. <111> and <100> type transport directions certainly have different values for aBlock spacings of atomic cores. The G, X, and L labels are a generic way to deal with this. (d) VM Ayres, ECE874, F12

  25. Two points before moving on to effective mass: • Kronig-Penney boundary conditions • Crystal momentum, the Uncertainty Principle and wavepackets VM Ayres, ECE874, F12

  26. Boundary conditions for Kronig-Penney model: Can you write these blurry boundary conditions without looking them up? VM Ayres, ECE874, F12

  27. Locate the boundaries: aKP + b = aBlock b aKP [transport direction p 56] -b 0 a a -b VM Ayres, ECE874, F12

  28. Locate the boundaries: into and out of the well. aKP + b = aBlock b aKP [transport direction p 56] -b 0 a a -b VM Ayres, ECE874, F12

  29. Boundary conditions for Kronig-Penney model, p. 57: Is the a in these equations aKP or aBl? VM Ayres, ECE874, F12

  30. Boundary conditions for Kronig-Penney model, p. 57: Is the a in these equations aKP or aBl? It is aKP. VM Ayres, ECE874, F12

  31. Two points before moving on to effective mass: • Kronig-Penney boundary conditions • Crystal momentum, the Uncertainty Principle and wavepackets VM Ayres, ECE874, F12

  32. VM Ayres, ECE874, F12

  33. Chp. 04: learn how to find the probability that an e- actually makes it into - “occupies” - a given energy level E. VM Ayres, ECE874, F12

  34. k2 k  wavenumber Chp. 02 VM Ayres, ECE874, F12

  35. Suppose U(x) is a Kronig-Penney model for a crystal. VM Ayres, ECE874, F12

  36. On E-axis: Allowed energy levels in a crystal, which an e- may occupy So a dispersion diagram is all about crystal stuff but there is an easy to understand connection between crystal energy levels E and e- ‘s occupying them. The confusion with momentum is that an e-’s real momentum is a particle not a wave property. Which brings us to the need for wavepackets. hbark = crystal momentum http://en.wikipedia.org/wiki/Crystal_momentum VM Ayres, ECE874, F12

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