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

2. 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. 3 We have defined the effective masses (m l * and m t *) in a valley minimum in Brillouin zone. We now want to define another type of effective mass in the whole Brillouin zone to account for all valley minima: DOS Effective Mass m* ds Here DOS denotes Density of States. States (defined by Pauli exclusion principle) can be thought of as available seats for electrons in conduction band as well as for holes in valence band. DOS

4. 4 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 DOS Regarding this point, textbooks would be helpful.

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

6. 6 1.Conduction Band GaAs: m dse * = m e * Silicon and Germanium: m dse * = g 2/3 (m l *m t * 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 m dsh * = ((m hh *) 3/2 + (m lh *) 3/2 ) 2/3 (Here for simplicity, we do not consider the Split-off band) 3-D Case

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

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

9. 9 n  N C exp(  C ) p  N V exp(  V ) N C = 2(m dse *k B T/2  ħ 2 ) 3/2 N V = 2(m dsh *k B T/2  ħ 2 ) 3/2 Effective density of states in the conduction band Effective density of states in the valence band  C = (E f – E C )/k B T  V = (E V – E f )/k B T Hole concentration Electron concentration Case of E V < E f < E C (Non-degenerate) Note: for E V < E f < E C, Fermi-Dirac distribution reduces to Boltzmann distribution.

10. 10 n  N C exp(  C ) p  N V exp(  V ) N C = 2(m dse *k B T/2  ħ 2 ) 3/2 N V = 2(m dsh *k B T/2  ħ 2 ) 3/2  C = (E f – E C )/k B T  V = (E V – E f )/k B T Case of E V < E f < E C (Non-degenerate) For intrinsic case where n = p, at least four statements can be drawn: E f is the intrinsic Fermi level E fi E fi is a function of the temperature T and the ratio of m dse * to m dsh * Corresponding n i (= n = p) is the intrinsic concentration n i is a function of the band gap (= E c - E v )

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

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

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