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Understanding Energy Minimization in Semiconductor Physics

This lecture by Prof. Virginia Ayres from Michigan State University delves into physical electronics, focusing on the energy minimization principles in semiconductors. Students explore the relationship between energy states and crystal directions, involving practical examples from materials like GaAs, Si, and Ge. The lecture covers equations relevant to k-space, energy valleys, and curvature in 3D representations. Through integration and analysis of energy-minimizing directions, students gain insights into the effective mass of electrons and holes, enhancing their understanding of semiconductor behavior in electronics.

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Understanding Energy Minimization in Semiconductor Physics

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

  2. Lecture 17, 25 Oct 12 VM Ayres, ECE874, F12

  3. VM Ayres, ECE874, F12

  4. VM Ayres, ECE874, F12

  5. VM Ayres, ECE874, F12

  6. VM Ayres, ECE874, F12

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

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

  9. VM Ayres, ECE874, F12

  10. Find [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). Final answer will depend ontime VM Ayres, ECE874, F12

  11. 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

  12. E – EV (eV) Same: truncate 1/2 L G X <111> <100> Picture taken from Ge, but same situation in GaAs in L direction VM Ayres, ECE874, F12

  13. VM Ayres, ECE874, F12

  14. VM Ayres, ECE874, F12

  15. Consider just the lowest energy and nearby: VM Ayres, ECE874, F12

  16. Goal: make these plausible: For GaAS For Si and Ge VM Ayres, ECE874, F12

  17. Consider just the lowest energy and nearby: GaAs: rectangular <100> directions are symmetric with diagonal <111> directions VM Ayres, ECE874, F12

  18. Equation of a sphere VM Ayres, ECE874, F12

  19. For Si, E-k is NOT symmetric in X and L: k1 = kz k3 = ky k2 = kx But X is symmetric across a face area VM Ayres, ECE874, F12

  20. Equation of an ellipsoid VM Ayres, ECE874, F12

  21. For Ge, E-k is also NOT symmetric in X and L, AND L is the minimum energy direction: Want this direction type to be the k1 direction with k2 and k3 defined to be orthogonal (transverse) to it. Equation of an ellipsoid VM Ayres, ECE874, F12

  22. Equation of an ellipsoid For Si and Ge: BUT: Ge k1 points in a diagonal type direction Si k1 points in a rectangular type direction VM Ayres, ECE874, F12

  23. Can show: P. 80: Can get ml*and mt* effective masses experimentally That means: can get an experimental measure of extent of k-space around the energy minima VM Ayres, ECE874, F12

  24. Use this in Chp. 04 too. VM Ayres, ECE874, F12

  25. (a) Confirm: http://en.wikipedia.org/wiki/Spheroid VM Ayres, ECE874, F12

  26. (a) VM Ayres, ECE874, F12

  27. (b) Conduction band minimum energy “valleys” VM Ayres, ECE874, F12

  28. (b) Temp not specified At 4K Does match ellipsoids as shown: Ge = long and skinny Si = not so long and not so skinny VM Ayres, ECE874, F12

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