Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University

1. DEE4521 Semiconductor Device Physics Lecture 3A: Density-of-States (DOS), Fermi-Dirac Statistics, and Fermi Level Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University October 1, 2013

2. What are States? • Pauli exclusion principle: No two electrons in a system can have the same set of quantum numbers. • Here, Quantum Numbers represent States.

3. DOS • We have defined the effective masses (ml* and mt*) in avalley minimum in Brillouin zone. • We now want to define another type of effective mass in the whole Brillouin zone toaccount for allvalley minima: DOS Effective Massm*ds • Here DOS denotes Density of States. • States(defined by Pauli exclusion principle) can be thought of as available seats forelectrons in conduction band as well as forholes in valence band.

4. DOS Ways to derive DOS and hence its DOS effective mass: • Solve Schrodinger equation in x-y-z space to find corresponding k solutions • Again apply the Pauli exclusion principle to these k solutions – spin up and spin down • Mathematically Transform an ellipsoidal energy surface to a sphere energy surface, particularly for Si and Ge Regarding this point, textbooks would be helpful.

5. 3-D Carriers S(E): DOS function, the number of states per unit energy per unit volume. mdse*: electron DOS effective mass, which carries the information about the DOS in conduction band mdsh*: hole DOS effective mass, which carries the information about the DOS in valence band

6. 3-D Case • Conduction Band • GaAs: mdse* = me* • Silicon and Germanium: mdse* = g2/3(ml*mt*2)1/3 where the degeneracy factor g is the number of ellipsoidal constant-energy surfaces lying within the Brillouin zone. For Si, g = 6; For Ge, g = 8/2 = 4. 2.Valence Band – Ge, Si, GaAs mdsh* = ((mhh*)3/2 + (mlh*)3/2)2/3 (Here for simplicity, we do not consider the Split-off band)

7. Fermi-Dirac Statistics Fermi-Dirac distribution function gives the probability of occupancy of an energy state E if the state exists. 1 - f(E): the probability of unfilled state E Ef: Fermi Level

8. Fermi level is related to one of laws of Nature: Conservation of Charge 2-13 Extrinsic case

9. Case of EV < Ef < EC (Non-degenerate) C = (Ef – EC)/kBT Electron concentration nNC exp(C) p  NV exp(V) V= (EV – Ef)/kBT Hole concentration Effective density of states in the conduction band NC = 2(mdse*kBT/2ħ2)3/2 NV = 2(mdsh*kBT/2ħ2)3/2 Effective density of states in the valence band Note: for EV < Ef< EC, Fermi-Dirac distribution reduces to Boltzmann distribution.

10. Case of EV < Ef < EC (Non-degenerate) C = (Ef – EC)/kBT nNC exp(C) p  NV exp(V) V= (EV – Ef)/kBT NC = 2(mdse*kBT/2ħ2)3/2 NV = 2(mdsh*kBT/2ħ2)3/2 • For intrinsic case where n = p, at least four statements can be drawn: • Ef is the intrinsic Fermi level Efi • Efi is a function of the temperature T and the ratio of mdse* to mdsh* • Corresponding ni (= n = p) is the intrinsic concentration • ni is a function of the band gap (= Ec- Ev)

13. (Continued from Lecture 2) Conduction-Band Electrons and Valence-Band Holes and Electrons Hole: Vacancy of Valence-Band Electron

14. No Electrons in Conduction Bands All Valence Bands are filled up.