1. 1 Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University September 18, 2014 DEE4521 Semiconductor Device Physics Lecture 2b: Density-of-States (DOS), Fermi-Dirac Statistics, and Fermi Level Density-of-States (DOS), Fermi-Dirac Statistics, and Fermi Level

2. 2 Extrinsic Semiconductors in Equilibrium Extrinsic Semiconductors in Equilibrium (open-circuited or short-circuited condition, for (open-circuited or short-circuited condition, for instance) instance) (Uniform and Non-uniform Doping) (Uniform and Non-uniform Doping)

3. 3 How to incorporate the equilibrium or quasi-equilibrium situation to device physics? Answer: Just keep Fermi level constant throughout the region of concern. (You will be able to prove this soon)

4. 4 2-5 Intrinsic Case (No Doping, No Impurities) Microscopic View n = p

5. 5 Silicon Crystal doped with phosphorus (donor) atoms. 2-6 n > p n-type

6. 6 Acceptors in a semiconductor An electron is excited from the valence band to the acceptor state, leaving behind a quasi-free hole. 2-8 p > n P-type

7. 7 2-13 Positioning of Fermi level can reveal the doping details

8. 8 2-14 n = n i exp((E f – E fi )/K B T) = N C exp((E f - E C )/K B T) p = n i exp((E fi – E f )/K B T) = N V exp((E V - E f )/K B T) pn = n i 2 for equilibrium n = p + N D + Ionized donor density E fi = (3/4)(K B T)ln(m dsh */m dse *) + (E c +E v )/2 extrinsic intrinsic n = p = n i

9. 9 2-15 E f itself reflects the charge conservation. n + N A - = p

10. 10 2-16 NA-NA- ND+ND+ n + N A - = p n = (N D + - N A - )+p N D + > N A - Compensation

11. 11 Electron distribution function n(E) 2-17 Evidence of DOF = 3

12. 12 Energy band-gap dependence of silicon on temperature 2-18

13. 13 EfEf

14. 14 2-19 Notice the doping concentrations illustrated, not so high!

15. 15 2-20 n i versus N D or N A Extrinsic temperature range for n i = N D (= N D + ) Full ionization of impurity Ionization energy < K B T

16. 16 Degenerate case (high doping)

17. 17 by Professor Pierret

18. 18 by Professor Pierret

19. 19 Band gap narrowing in heavily doped semiconductors?? -Will be clarified in the minority carrier injection experiment in a bipolar transistor (in subsequent bipolar chapter). -So, let us skip it now until going into bipolar chapter.

20. 20 Non-uniform Doping

21. 21 4-2 Non-uniformly doped semiconductor Only for doping with a non-uniform distribution can the Einstein relationship be derived.

22. 22 4-8

23. 23 Built-in Fields in Non-uniform Semiconductors 4-9