lecture8a

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SEMICONDUCTORSSEMICONDUCTORSSEMICONDUCTORSSEMICONDUCTORS

Diamond structureDiamond structureDiamond structureDiamond structure

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Gallium Arsenide structureGallium Arsenide structureGallium Arsenide structureGallium Arsenide structure

GalliumGalliumGalliumGallium is on a fcc latticeis on a fcc latticeis on a fcc latticeis on a fcc latticeArsenideArsenideArsenideArsenideArsenideArsenideArsenideArsenide is interstitialis interstitialis interstitialis interstitial

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Fermi surface for GaAsFermi surface for GaAsFermi surface for GaAsFermi surface for GaAs

Energy band in a semiconductorEnergy band in a semiconductorEnergy band in a semiconductorEnergy band in a semiconductor

Electron-hole pair

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Band structure of GaAsBand structure of GaAsBand structure of GaAsBand structure of GaAs

FourFourFourFour----band diagramband diagramband diagramband diagram

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TB TB TB TB point of viewpoint of viewpoint of viewpoint of view

Band structure of Si and GaAsBand structure of Si and GaAsBand structure of Si and GaAsBand structure of Si and GaAs

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Indirect or direct Indirect or direct Indirect or direct Indirect or direct adsoportionadsoportionadsoportionadsoportion

Indirect or direct gap?Indirect or direct gap?Indirect or direct gap?Indirect or direct gap?

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Energy band in a semiconductorEnergy band in a semiconductorEnergy band in a semiconductorEnergy band in a semiconductor

Electrons and holesElectrons and holesElectrons and holesElectrons and holes

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Carrier densitiesCarrier densitiesCarrier densitiesCarrier densities

Let be n(Let be n(Let be n(Let be n() the electron density of states per ) the electron density of states per ) the electron density of states per ) the electron density of states per unit energy. It is given by the product of unit energy. It is given by the product of unit energy. It is given by the product of unit energy. It is given by the product of the density of states gthe density of states gthe density of states gthe density of states gelelelel(((() times the Fermi) times the Fermi) times the Fermi) times the Fermi----Dirac distribution f(Dirac distribution f(Dirac distribution f(Dirac distribution f(): ): ): ):

Let be p(Let be p(Let be p(Let be p() the hole density of states per unit energy. It is given ) the hole density of states per unit energy. It is given ) the hole density of states per unit energy. It is given ) the hole density of states per unit energy. It is given by the product of the density of states by the product of the density of states by the product of the density of states by the product of the density of states gggghhhh(((() in the valence ) in the valence ) in the valence ) in the valence band times the Fermiband times the Fermiband times the Fermiband times the Fermi----Dirac distribution (1Dirac distribution (1Dirac distribution (1Dirac distribution (1----f(f(f(f()): )): )): )):

Electron densityElectron densityElectron densityElectron density

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Carrier Carrier Carrier Carrier densitiesdensitiesdensitiesdensities

FermiFermiFermiFermi----Dirac distribution at Dirac distribution at Dirac distribution at Dirac distribution at T=0K. T=0K. T=0K. T=0K.

In blue and in black tails of In blue and in black tails of In blue and in black tails of In blue and in black tails of FermiFermiFermiFermi----Dirac distribution at Dirac distribution at Dirac distribution at Dirac distribution at finite temperatures finite temperatures finite temperatures finite temperatures TTTTblackblackblackblack

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Carrier densitiesCarrier densitiesCarrier densitiesCarrier densities

( )

( )( )

pi

pi

pi

=

=

=

VB3

23

*h

h

CB3

23

*el

el

3

23

el

Eh

m28)(g

Eh

m28)(g

h

m28)(g

EEEEgapgapgapgap := E:= E:= E:= ECBCBCBCB----EEEEVBVBVBVB

Number of electronsNumber of electronsNumber of electronsNumber of electrons

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Number of holesNumber of holesNumber of holesNumber of holes

Si band structureSi band structureSi band structureSi band structure

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Fermi integral(s)Fermi integral(s)Fermi integral(s)Fermi integral(s)

For a nonFor a nonFor a nonFor a non----degenerate semiconductor, in the limit of low T meaning degenerate semiconductor, in the limit of low T meaning degenerate semiconductor, in the limit of low T meaning degenerate semiconductor, in the limit of low T meaning kkkkBBBBTTTT

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NonNonNonNon----degenerate semiconductorsdegenerate semiconductorsdegenerate semiconductorsdegenerate semiconductorsNonNonNonNon----degenerate semiconductors are defined as degenerate semiconductors are defined as degenerate semiconductors are defined as degenerate semiconductors are defined as semiconductors for which the Fermi energy semiconductors for which the Fermi energy semiconductors for which the Fermi energy semiconductors for which the Fermi energy FFFF is at east is at east is at east is at east 3333kkkkBBBBT T T T away from either band edge.away from either band edge.away from either band edge.away from either band edge.

In this approx., the FermiIn this approx., the FermiIn this approx., the FermiIn this approx., the Fermi----Dirac distribution could be Dirac distribution could be Dirac distribution could be Dirac distribution could be replaced by the Maxwellreplaced by the Maxwellreplaced by the Maxwellreplaced by the Maxwell----Boltzmann, given by:Boltzmann, given by:Boltzmann, given by:Boltzmann, given by:

Effective carrier densities Effective carrier densities Effective carrier densities Effective carrier densities

ECB

EVB

For nonFor nonFor nonFor non----degenerate semiconductorsdegenerate semiconductorsdegenerate semiconductorsdegenerate semiconductors, the , the , the , the FermiFermiFermiFermi----Dirac could be approximated by the Dirac could be approximated by the Dirac could be approximated by the Dirac could be approximated by the Maxwell distribution:Maxwell distribution:Maxwell distribution:Maxwell distribution:

Thus the electron carrier density is Thus the electron carrier density is Thus the electron carrier density is Thus the electron carrier density is

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Effective carrier densities Effective carrier densities Effective carrier densities Effective carrier densities

ECB

EVB

For nonFor nonFor nonFor non----degenerate semiconductors, the degenerate semiconductors, the degenerate semiconductors, the degenerate semiconductors, the FermiFermiFermiFermi----Dirac could be approximated by the Dirac could be approximated by the Dirac could be approximated by the Dirac could be approximated by the Maxwell distribution:Maxwell distribution:Maxwell distribution:Maxwell distribution:

Thus, the hole carrier density is: Thus, the hole carrier density is: Thus, the hole carrier density is: Thus, the hole carrier density is:

Mass action lawMass action lawMass action lawMass action law

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Fermi energy for a semiconductorFermi energy for a semiconductorFermi energy for a semiconductorFermi energy for a semiconductor

Keep in mindKeep in mindKeep in mindKeep in mind