Understanding Hall Effect in Semiconductor Devices

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

2. Lecture 14, 10 Feb 14 • Hw 04: FRI: Pr. 2.07 • Chp. 01 – Chp. 02 • Experimental measurements for concentration: • Hall effect – Chp. 01: material • C-V – Chp. 02: pn junction VM Ayres, ECE875, S14

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4. Igen = ? OR Irec = ? Which: are you in forward or reverse bias? What happens to the depletion region WD? VM Ayres, ECE875, S14

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6. = everything that’s left in U In Pr. 2.07: tg is given. VM Ayres, ECE875, S14

7. Reminder: Lec 13: When trying to turn a pn junction OFF, a substantial generation current makes this difficult Similarly, when trying to turn a pn junction ON, a substantial recombination current makes this difficult VM Ayres, ECE875, S14

8. Lecture 14, 10 Feb 14 • Hw 04: FRI: Pr. 2.07 • Chp. 01 – Chp. 02 • Experimental measurements for concentration: • Hall effect – Chp. 01: material • C-V – Chp. 02: pn junction VM Ayres, ECE875, S14

9. Add a Magnetic field B to a doped semiconductor with a current flowingOperator facing same direction as the historic current Hall Effect: • F = q (v x B) Consider majority holes: F = +e-(+vx x Bz) F+y = |e- vx Bz| on holes to RHS Displaced electrons go to LHS VAB = positive • F = q (v x B) Consider majority electrons: F = -e-(-vx x B F+y = |e- vx Bz| on electrons to RHS Displaced holes go to LHS VAB = negative VM Ayres, ECE875, S14

10. Hall Effect: no scattering: VAB p is majority carrier: + - I positive positive n is majority carrier: - + I negative negative VM Ayres, ECE875, S14

11. Hall Effect: no scattering: VAB p is majority carrier: + - I positive positive n is majority carrier: - + I negative negative VM Ayres, ECE875, S14

12. Hall Effect: ECE 875: with scattering: VAB p is majority carrier: + - I positive positive n is majority carrier: - + I negative negative VM Ayres, ECE875, S14

13. tm: time between scattering events Called the mean free lifetime, also called the momentum relaxation time The mean free lifetime depends on the energy the electron has: Mean free lifetimealso depends on the type of scatterer VM Ayres, ECE875, S14

14. Mean free lifetime can be equivalently described as a mean free length (momentum relaxation length): l also called lm and lm in Sze VM Ayres, ECE875, S14

15. Generally: Low temp T: impurity scattering: ND+, NA-: s = ½ High temp T: phonon scattering: s = 3/2 (further info in Chp. 01 eq’s (49) and (50)) VM Ayres, ECE875, S14

16. Example: choose the semiconductor with a spherical constant energy surface: Ge, Si, or GaAs VM Ayres, ECE875, S14

17. Answer: choose the semiconductor with a spherical constant energy surface: Ge, Si, or GaAs VM Ayres, ECE875, S14

18. <tm> , <tm2> depend on the definition of average < > : Stated without proof: eq’n (72): For a Boltzmann distribution of carriers in a non-degenerate semiconductor: (Note: normalization: Pr. 10: KE) VM Ayres, ECE875, S14

19. Hall Effect: evaluated for you: All you need to know is: s where: VM Ayres, ECE875, S14

20. Hall mobility mH from Hall factor rH: Related to: VM Ayres, ECE875, S14

21. mn and mp:

22. mn and mp:

23. Example: relate eq’n (54) to the result of Pr. 1.10:

24. Answer:

25. Example: VM Ayres, ECE875, S14

26. Example: Carrier density = ? Mobility = ? VM Ayres, ECE875, S14

27. Example: RH not Hall factor rH Carrier density = n OR p = ? Mobility = ? VM Ayres, ECE875, S14

28. Example: RH not Hall factor rH Carrier density = n since RH = negative and only one type of carrier is present Mobility = ? VM Ayres, ECE875, S14

29. Example: Carrier density = n since RH = negative and only one type of carrier is present Mobility = mn OR mHall VM Ayres, ECE875, S14

30. Example: Note: different s Carrier density = n since RH = negative and only one type of carrier is present Mobility = mH VM Ayres, ECE875, S14

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33. 2 VM Ayres, ECE875, S14

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36. Scattering also depends on the type of scatterer: Low temp T: ND+, NA-: s = ½ High temp T: phonons: s = 3/2 VM Ayres, ECE875, S14

37. Phonon model: 1D vibrational modes for a linear chain with unequal masses: 1D: m1 m2 Frequency n±: Symmetric and anti-symmetric motion: ± Low frequency (acoustic) and high frequency (optical) solution Equation of motion from F = ma is variation on a harmonic oscillator with multiple solution given by integer kph = 0, 1, 2, …

38. Phonon model: 1D vibrational modes for a linear chain with unequal masses: 1D: m1 m2 Frequency n±: Symmetric and anti-symmetric motion: ± Low frequency (acoustic) and high frequency (optical) solution Equation of motion from F = ma is variation on a harmonic oscillator with multiple solution given by integer kph = 0, 1, 2, …

39. High field effects: Drift + Diffusion Current Densities

40. High field effects: ECE 474:

41. Curved Zero slope Linear slope

42. E to e- then e- to acoustic phonons Interaction with acoustic phonons Interaction with optical phonons

43. High field effects: E to e- then e- to acoustic phonons Feeding energy to acoustic phonons => more interactions with acoustic phonons Interactions with phonons become significant: when m0E becomes comparable with speed of sound cs

44. High field effects: Electron interactions with optical phonons Multiple mechanisms for energy feeding and electron-phonon interactions possible, not simple balance Stated without proof: Empirical relationship for vd for all 3 regimes:

46. Phonons: Stated without proof: 3D: The total number of acoustic modes = dimension X number of atoms per primitive cell Example: Si: Dimension: 3D Number of atoms per primitive cell: Number of acoustic modes = 6 P. 50: “three acoustic and three optical”: degeneracy

47. Phonons Degeneracy in these compact diagrams too: <100> k = p/a k = 0