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

2. VM Ayres, ECE875, S14

3. Revised Lecture 06: Slide 19:
Sze:
VM Ayres, ECE875, S14

4. Revised Lecture 06: Slide 20:
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5. Revised Lecture 06: Slide 19:
Sze:
Pierret: average effective mass definition includes MC
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6. VM Ayres, ECE875, S14

7. VM Ayres, ECE875, S14

8. } Chp. 01 Lecture 07, 24 Jan 14 Energy levels: E-k Effective mass mij*
vgroup
Density of States
Concentrations
Degenerate
Effect of temperature
Contributed by traps }
VM Ayres, ECE875, S14

9. Carrier concentration as a function of temperature:
F(E) = a probability = a number between 0 and 1 (between 0% to 100%)
Probability of what? Probability that an electron occupies an allowed energy level which has energy value E(k). The energy of the electron matches what is allowed by its physical environment.
VM Ayres, ECE875, S14

10. 1. Plot of F(E), temperature dependence shown
For electrons with spin up/spin down, correct expression for probability is:
1. Plot of F(E), temperature dependence shown
At E = EF,
F(E) = ½ = 50%
E – EF (eV) = 0 at E = EF
Note for slide 12: Could also make the x-axis: E, not E - EF
2. For concentration, consider where EF is relative to EC and EV
VM Ayres, ECE875, S14

11. Plot of DOS N(E): VM Ayres, ECE875, S14
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12. VM Ayres, ECE875, S14

13. - 3. To carry out the integral: Useful approximations:
Then: must use the denominator:
Use the “hot” limit in ECE 474 & 874 & 875: Si electrons -
Use the “cold” limit in ECE
Nanoelectronics
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14. - Need E > EF : nondegenerate doping
E = EC or above is OK and EF is in usual position below EC + -
Need EF > E
Therefore:
“Hot and cold” depend on ± |E – EF | / kT, not on whether T is large or small.
Get there by heavy (degenerate) doping (Sze: 3D Si electronics) OR by naturally high carrier concentrations (2D, 1D nanoelectronics)
VM Ayres, ECE875, S14

15. Approximation/limits on F(E): Hot limit:-
= F(E)
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16. Approximation/limits on F(E): Cold limit:1
(step function at E = EF)
-| |
= F(E)
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17. As the Hot limit approaches the Cold limit: “within the degenerate limit”F E i E V
Use:
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18. Carrier concentration as a function of temperature:
Identify the limits
Carry out the integration
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19. “hot” approximation of Eq’n (16)
Example: Concentration of conduction band electrons for a nondegenerate semiconductor: n:
3D: Eq’n (14)
“hot” approximation of Eq’n (16)
1/3
(000) Si
GaAs
VM Ayres, ECE875, S14 Ge

20. “hot” approximation of Eq’n (16)
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)
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21. 1st : Deal with E – EF versus E – EC:
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22. 2nd : Deal with E – EC versus dE:
Let:
Then:
MAKE SURE you remember to change the limits of integration:
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23. 2nd : Deal with E – EC versus dE:
Practically speaking, any upper energy level that takes about 0.4 eV to reach from the bottom of EC seems to require DE “infinite” amount of energy to an e-:
Chance of occupancy = really low
DE ~ 0.4 eV
Therefore can simplify the math:
VM Ayres, ECE875, S14

24. de
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25. VM Ayres, ECE875, S14

26. Always do the easy one first: denominator:
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27. Then do the numerator:
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28. Done! For 3D non-degenerate semiconductors:MC
The part in the bracket is called NC: the effective density of states at the conduction band edge.
VM Ayres, ECE875, S14

29. 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
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30. F(E) considered as a distribution:
F(E) = a distribution = a plot that show frequency of occurrence
Frequency of occurrence of what?
Consider as a histogram. x-axis: electron energy “bins”
y-axis: “property”: probability.
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31. VM Ayres, ECE875, S14

32. Example: Sze Pr. 1.10:
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33. “hot” approximation of Eq’n (16) 3D: Eq’n (14)
Average Kinetic Energy
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