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Understanding Electron Concentration in Nondegenerate Semiconductors

This lecture from ECE 875, taught by Prof. Virginia Ayres at Michigan State University, explores the concepts of electron concentrations in nondegenerate and degenerate semiconductors. Key topics include the effects of temperature, the importance of effective density of states in defining conduction and valence bands, and the application of the "hot" approximation. Students will learn how to calculate electron concentrations using established equations, examine the role of traps, and visualize electron dynamics through kinetic energy considerations.

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Understanding Electron Concentration in Nondegenerate Semiconductors

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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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