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

2 1. To count electrons or holes, it is better started from the Energy point of view. 2. DOS and Fermi Statistics and Level are Energy related terminologies.

3 Let us focus on those of textbook contents involved: pp. 71 – 102; pp. 71 – 102;and start from the (potential) energy band diagram (Remember: you should have got it from a slide example and a conductor example).

4 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. States can be thought of as available seats for electrons in conduction band, as well as for holes in valence band.

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

6 Fermi-Dirac distribution function delivers the probability of filling an existing state at energy E. Fermi-Dirac Statistics 1 - f(E): the probability of unfilled state E E f : Fermi Level

7 Key Concepts: 1. Fermi Level only exists in semiconductors, metals, and around insulators’ edges, NOT bulk insulators. 2. Fermi-Dirac Statistics (and hence Fermi level) only applies to mobile electrons, mobile holes, and immobile (valence) electrons, NOT phonons, plasmons, or (ionized) impurity level. 3. Fermi level must be first constructed in the equilibrium or quasi- equilibrium condition and in the neutral or quasi-neutral region. 4. Fermi level is the only direct way with which our EE people can go into device.

Fermi level is related to one of laws of Nature: Conservation of Charge Extrinsic (doping) case Positioning of Fermi level can reveal the doping details.

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 Evidence of DOF = 3 Electron distribution function n(E)

11 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 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 )

12 DOF (degree of freedom): d

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