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Prof. Ming-Jer ChenProf. Ming-Jer Chen

Department of Electronics EngineeringDepartment of Electronics Engineering

National Chiao-Tung UniversityNational Chiao-Tung University

October 1, 2013October 1, 2013

DEE4521 Semiconductor Device PhysicsDEE4521 Semiconductor Device Physics

Lecture 3A:Lecture 3A: Density-of-States (DOS), Fermi-Dirac Statistics, and Fermi LevelDensity-of-States (DOS), Fermi-Dirac Statistics, and Fermi Level

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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.

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• We have defined the effective masses (ml* and mt*) 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

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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.

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Cdse EE

mES 2/3

2

*

2)

2(

2

1)(

EEm

ES Vdsh 2/32

*

2)

2(

2

1)(

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 bandmdsh*: hole DOS effective mass, which carries the information about the DOS in valence band

3-D Carriers

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1. 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)

3-D Case

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Fermi-Dirac distribution function gives the probability of occupancy of an energy state E if the state exists.

( )/

1( )

1 f BE E k Tf Ee

Fermi-Dirac Statistics

1 - f(E): the probability of unfilled state EEf: Fermi Level

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2-13

Fermi level is related to one of laws of Nature:Conservation of Charge

Extrinsic case

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CE

dEEfESn )()(

VE

dEEfESp ))(1)((

n NC exp(C)

p NV exp(V)

NC = 2(mdse*kBT/2ħ2)3/2

NV = 2(mdsh*kBT/2ħ2)3/2

Effective density of statesin the conduction band

Effective density of statesin the valence band

C = (Ef – EC)/kBT

V= (EV – Ef)/kBTHole concentration

Electron concentration

Case of EV < Ef < EC (Non-degenerate)

Note: for EV < Ef < EC, Fermi-Dirac distribution reduces to Boltzmann distribution.

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n NC exp(C)

p NV exp(V)

NC = 2(mdse*kBT/2ħ2)3/2

NV = 2(mdsh*kBT/2ħ2)3/2

C = (Ef – EC)/kBT

V= (EV – Ef)/kBT

Case of EV < Ef < EC (Non-degenerate)

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)

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Conduction-Band Electrons andValence-Band Holes and Electrons

Hole: Vacancy of Valence-Band Electron

(Continued from Lecture 2)

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No Electrons in Conduction Bands

All Valence Bands are filled up.

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