1 Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University October 1, 2013 DEE4521 Semiconductor Device Physics Lecture 3B: 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 (Uniform and Non-uniform Doping) (Uniform and Non-uniform Doping)

3 Uniform Doping We first focus on Non-Degenerate semiconductors, the Case of low and moderate doping of less than cm -3.

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

5 Silicon Crystal doped with phosphorus (donor) atoms. 2-6 One typical method to dope or introduce impurities: High-energy ion implant at room temperature, followed by High temperature (  1000 o C) annealing (to eliminate the defects and to place impurities on the lattice positions correctly) n > p

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

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

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

15 How to deal with Degenerate case (very high doping)?

16 by Professor Pierret

17 by Professor Pierret

18 EfEf

19 Non-uniform Doping Case

Nonuniformly doped semiconductor Only for doping with non-uniform distribution can Einstein relationship be derived.

21 4-8

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