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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 16, 23 Oct 12. Effective mass: How: practical discussion:. Reminder: how you got the E-k curves: Kronig-Penney model allowed energy levels, Chp. 03:. LHS.

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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 16, 23 Oct 12 VM Ayres, ECE874, F12

  3. Effective mass: How: practical discussion: VM Ayres, ECE874, F12

  4. Reminder: how you got the E-k curves:Kronig-Penney model allowed energy levels, Chp. 03: LHS RHS 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

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

  6. (b) VM Ayres, ECE874, F12

  7. (b) VM Ayres, ECE874, F12

  8. Get: the E-k curves. Matlab can do numerical derivatives Note that the effective mass m* isn’t a single number. Note also that a + b = aBl varies depending on what direction you move in, so there are more curves than are on this single ± direction chart. VM Ayres, ECE874, F12

  9. Get: the E-k curves. Region of biggest change of tangent = greatest curvature: the parabolas shown. Example problem: Which band has the sharpest curvature d2E/dk2? Which band has the lightest effective mass? Which band has the heaviest effective mass? Where in k-space, for both? VM Ayres, ECE874, F12

  10. Get: the E-k curves. Region of biggest change of tangent = greatest curvature: the parabolas shown. Example problem: Which band has the sharpest curvature d2E/dk2? Band 4 Which band has the lightest effective mass? Which band has the heaviest effective mass? Band 1: broadest = least curvature divide by smallest number = heaviest m* Where in k-space, for both? At k= 0 called the G point VM Ayres, ECE874, F12

  11. VM Ayres, ECE874, F12

  12. Where in k-space, for both? VM Ayres, ECE874, F12

  13. Where in k-space, for both? m*A at G k = 0 m*B at about ½ way between G and X in [100] direction: k = 0 VM Ayres, ECE874, F12

  14. a + b = aBl aBl for [100] = aLC k = p/aBl = p/aLC at end of Zone 1 This is X for [100] VM Ayres, ECE874, F12

  15. Where in k-space, for both? m*A at G k = 0 m*B at about ½ way between G and X in [100] direction at k = p/2 aLC k = 0 VM Ayres, ECE874, F12

  16. Assume T = 300K and it doesn’t change Ec = Egap = constant at a given T Hint: compare the answers for b = 0 and b ≠ 0 in (a) VM Ayres, ECE874, F12

  17. Pick correct curve: VM Ayres, ECE874, F12

  18. Pick conduction or valence bands:: E – Ec (eV) VM Ayres, ECE874, F12

  19. Pick conduction minima. Where in k-space are they? E – EV (eV) L G X <111> <100> VM Ayres, ECE874, F12

  20. Pick conduction minima. Where in k-space are they? G at k = 0 L at k = p/aBl for <111> Could work out the aBl distance between atomic cores in a <111> direction if needed. Not needed to finish answering the question. E – EV (eV) L G X <111> <100> VM Ayres, ECE874, F12

  21. Go back to here: Note that the effective mass m* isn’t a single number. Note also that a + b = aBl varies depending on what direction you move in, so there are more curves than are on this single ± direction chart. VM Ayres, ECE874, F12

  22. VM Ayres, ECE874, F12

  23. VM Ayres, ECE874, F12

  24. VM Ayres, ECE874, F12

  25. VM Ayres, ECE874, F12

  26. (From practical to fundamental!) VM Ayres, ECE874, F12

  27. In 3 D: VM Ayres, ECE874, F12

  28. Write this in 2D: all three parts. Integrate a -> v -> r. Vector r (t) is the direction. The final answer contains time t. VM Ayres, ECE874, F12

  29. Start with [m*ij] Then F = qE Then a = dv/dt for dvx/dt and dvy/dt Integrate with respect to time, 2x’s, to get x(t) and y(t). VM Ayres, ECE874, F12

  30. k = 0 VM Ayres, ECE874, F12

  31. Region of biggest change of tangent = greatest curvature: the parabolas shown. 1D: Any one of these parabolas could be modelled as: VM Ayres, ECE874, F12

  32. Region of biggest change of tangent = greatest curvature: the parabolas shown. 3D: <111> + <100> E – EV (eV) L G X <111> <100> For any of these parabolas: There’s a major axis but also two minor ones VM Ayres, ECE874, F12

  33. E – EV (eV) Same: truncate 1/2 L G X <111> <100> VM Ayres, ECE874, F12

  34. k = 0 VM Ayres, ECE874, F12

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