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

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

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

Positioning of Fermi level can reveal the doping details

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

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

NA-NA- ND+ND+ n + N A - = p n = (N D + - N A - )+p N D + > N A - Compensation

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

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

13 EfEf

Notice the doping concentrations illustrated, not so high!

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 Degenerate case (high doping)

17 by Professor Pierret

18 by Professor Pierret

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 Non-uniform Doping

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

22 4-8

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