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ECE 875: Electronic Devices

ECE 875: Electronic Devices. Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu. Lecture 08, 27 Jan 14. Chp. 01 Concentrations Degenerate Nondegenerate Effect of temperature Contributed by traps. }.

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ECE 875: Electronic Devices

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  1. ECE 875:Electronic Devices Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu

  2. Lecture 08, 27 Jan 14 • Chp. 01 • Concentrations • Degenerate • Nondegenerate Effect of temperature • Contributed by traps } VM Ayres, ECE875, S14

  3. Example:Concentration of conduction band electrons for a nondegenerate semiconductor: n: 3D: Eq’n (14) “hot” approximation of Eq’n (16) Three different variables (NEVER ignore this) VM Ayres, ECE875, S14

  4. Answer:Concentration of conduction band electrons for a nondegenerate semiconductor: n: MC NC The effective density of states at the conduction band edge. VM Ayres, ECE875, S14

  5. Answer:Concentration of conduction band electrons for a nondegenerate semiconductor: n: Nondegenerate: EC is above EF: Sze eq’n (21) Use Appendix G at 300K for NC and n ≈ ND when fully ionised VM Ayres, ECE875, S14

  6. Lecture 07: Would get a similar result for holes: This part is called NV: the effective density of states at the valence band edge. Typically valence bands are symmetric about G: MV = 1 VM Ayres, ECE875, S14

  7. Similar result for holes:Concentration of valence band holes for a nondegenerate semiconductor: p: Nondegenerate: EC is above EF: Sze eq’n (23) Use Appendix G at 300K for NV and p ≈ NA when fully ionised VM Ayres, ECE875, S14

  8. HW03: Pr 1.10: Shown: kinetic energies of e- in minimum energy parabolas: KE  E > EC. Therefore: generic definition of KE as: KE = E - EC VM Ayres, ECE875, S14

  9. HW03: Pr 1.10: Define: Average Kinetic Energy Single band assumption VM Ayres, ECE875, S14

  10. HW03: Pr 1.10: “hot” approximation of Eq’n (16) 3D: Eq’n (14) Average Kinetic Energy Single band assumption VM Ayres, ECE875, S14

  11. HW03: Pr 1.10: “hot” approximation of Eq’n (16) 3D: Eq’n (14) Average Kinetic Energy Equation 14: Single band definition VM Ayres, ECE875, S14

  12. Considerations: VM Ayres, ECE875, S14

  13. Therefore: Single band assumption: means: VM Ayres, ECE875, S14

  14. Therefore: Use a Single band assumption in HW03: Pr 1.10: “hot” approximation of Eq’n (16) 3D: Eq’n (14) Start: Average Kinetic Energy Finish: Average Kinetic Energy VM Ayres, ECE875, S14

  15. Reference: http://en.wikipedia.org/wiki/Gamma_function#Integration_problems Some commonly used gamma functions: n is always a positive whole number VM Ayres, ECE875, S14

  16. Because nondegenerate: used the Hot limit: E C E F E i E V - = F(E) VM Ayres, ECE875, S14

  17. Consider: as the Hot limit approaches the Cold limit:“within the degenerate limit” E C E F E i E V Use: VM Ayres, ECE875, S14

  18. Will find: useful universal graph: from n: Dotted: nondegenerate Solid: within the degenerate limit y-axis: Fermi-Dirac integral: good for any semiconductor x-axis: how much energy do e-s need: (EF – EC) versus how much energy can they get: kT VM Ayres, ECE875, S14

  19. Concentration of conduction band electrons for a semiconductor within the degenerate limit: n: 3D: Eq’n (14) Three different variables (NEVER ignore this) VM Ayres, ECE875, S14

  20. Part of strategy: pull all semiconductor-specific info into NC. To get NC: VM Ayres, ECE875, S14

  21. Next: put the integrand into one single variable: VM Ayres, ECE875, S14

  22. Next: put the integrand into one single variable: Therefore have: And have: VM Ayres, ECE875, S14

  23. Next: put the integrand into one single variable: Change dE: Remember to also change the limits to hbottom and htop: VM Ayres, ECE875, S14

  24. Now have: Next: write “Factor” in terms of NC: VM Ayres, ECE875, S14

  25. Write “Factor” in terms of NC: Compare: VM Ayres, ECE875, S14

  26. Write “Factor” in terms of NC: VM Ayres, ECE875, S14

  27. F1/2(hF) No closed form solution but correctly set up for numerical integration VM Ayres, ECE875, S14

  28. Note: • hF = (EF - EC)/kT is semiconductor-specific • F1/2(hF) is semiconductor-specific • But: a plot of F1/2(hF) versus hF is universal Could just as easily write this as F1/2(x) versus x VM Ayres, ECE875, S14

  29. Recall: on Slide 5 for a nondegenerate semiconductor: n: 3D: Eq’n (14) “hot” approximation of Eq’n (16) F1/2(hF) VM Ayres, ECE875, S14

  30. Useful universal graph: Dotted: nondegenerate Solid: within the degenerate limit y-axis: Fermi-Dirac integral: good for any semiconductor x-axis: how much energy do e-s need: (EF – EC) versus how much energy can they get: kT VM Ayres, ECE875, S14

  31. VM Ayres, ECE875, S14

  32. Why useful: one reason: Around -1.0 Starts to diverge -0.35: ECE 874 definition of “within the degenerate limit” Shows where hot limit becomes the “within the degenerate limit” EC EF Ei EV VM Ayres, ECE875, S14

  33. Why useful: another reason: F(hF)1/2 integral is universal: can read numerical solution value off this graph for any semiconductor Example: p.18 Sze: What is the concentration n for any semiconductor when EF coincides with EC? VM Ayres, ECE875, S14

  34. Why useful: another reason: Answer: Degenerate EF = EC => hF = 0 Read off the F1/2(hF) integral value at hF = 0 ≈ 0.6 VM Ayres, ECE875, S14 Appendix G

  35. Example:What is the concentration of conduction band electrons for degenerately doped GaAs at room temperature 300K when EF – EC = +0.9 kT? EF 0.9 kT EC Ei EV VM Ayres, ECE875, S14

  36. Answer: VM Ayres, ECE875, S14

  37. For degenerately doped semiconductors (Sze: “degenerate semiconductors”): the relative Fermi level is given by the following approximate expressions: VM Ayres, ECE875, S14

  38. Compare: Sze eq’ns (21) and (23): for nondegenerate: Compare with degenerate: VM Ayres, ECE875, S14

  39. Lecture 08, 27 Jan 14 • Chp. 01 • Concentrations • Degenerate • Nondegenerate Effect of temperature • Contributed by traps } VM Ayres, ECE875, S14

  40. Nondegenerate: will show: this is the Temperature dependence of intrinsic concentrations ni = pi ECE 474 VM Ayres, ECE875, S14

  41. Intrinsic: n = pIntrinsic: EF =Ei = Egap/2 Correct definition of intrinsic: Set concentration of e- and holes equal: For nondegenerate: = VM Ayres, ECE875, S14

  42. Solve for EF: EF for n = p is given the special name Ei VM Ayres, ECE875, S14

  43. Substitute EF = Ei into expression for n and p. n and p when EF = Ei are given name: intrinsic: ni and pi ni = pi = ni = pi: VM Ayres, ECE875, S14

  44. Substitute EF = Ei into expression for n and p. n and p when EF = Ei are given name: intrinsic: ni and pi ni = pi = ni = pi: VM Ayres, ECE875, S14 Units of 4.9 x 1015 = ? = cm-3 K-3/2

  45. Plot: ni versus T: ni Note: temperature is not very low 1018 106 VM Ayres, ECE875, S14

  46. Dotted line is same relationship for ni as in the previous picture.However: this is doped Si: < liquid N2 1017 When temperature T = high, most electrons in concentration ni come from Si bonds not from dopants 1013 VM Ayres, ECE875, S14

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