Variation of Energy Bands with Alloy Compositionfolk.uio.no/ravi/cutn/semiphy/7.l8_carrier_const.pdfThe atomic mass of silicon is 28.1 g which contains Avagadro’s number of atoms.

P.Ravindran, PHY02E Semiconductor Physics, 31 January 2013: Carriers in Semiconductor1

Variation of Energy Bands with Alloy Composition

X

L

1.43eVk

E

0.3eV

AlxGa1-

xAs

2.16eV

AlAsGaAsX

E

1.4

2.0

1.8

1.6

2.2

2.4

2.6

2.8

3.0

0 0.2 0.4 0.6 0.8 1

X

L

X

L


Carriers in intrinsic Semiconductors

Ec

Ev

Eg0ºK3ºK2ºK4ºK5ºK1ºK6ºK7ºK8ºK9ºK10ºK11ºK12ºK13ºK14ºK300ºK15ºK16ºK17ºK18ºK19ºK20ºK

Electron Hole PairE H P

P.Ravindran, PHY02E Semiconductor Physics, 31 January 2013: Carriers in Semiconductor

Effective Mass Example:

Find the (E,k) relationship for a free electron and relate it to the electron mass.

E

k

khmvp

222

2

22

1

2

1k

m

h

m

pmvE

m

h

dk

Ed 2

2

2

Most energy bands are close to parabolic at their minima (for conduction bands) or maxima (for valence bands).


Intrinsic Material

• A perfect semiconductor crystal with no impurities or

lattice defects is called an Intrinsic semiconductor.

• In such material there are no charge carriers at 0ºK,

since the valence band is filled with electrons

and the conduction band is empty.


Intrinsic Material

Si

Eg

h+

e-

n=p=ni


Extrinsic Material

In addition to the intrinsic carriers generated thermally, it is possible to create carriers in semiconductors by purposely introducing impurities into the crystal. This process, called doping, is the most common technique for varying the conductivity of semiconductors.

When a crystal is doped such that the equilibrium carrier concentrations n0and p0 are different from the intrinsic carrier concentration ni , the material

is said to be extrinsic.


Extrinsic Material (n-type)

0ºK3ºK2ºK4ºK5ºK1ºK6ºK7ºK8ºK9ºK10ºK11ºK12ºK13ºK14ºK50ºK15ºK16ºK17ºK18ºK19ºK20ºK

Ec

Ev

Ed

Donor

V

P

As

Sb


Extrinsic Material (p-type)

ш

B

Al

Ga

In

0ºK3ºK2ºK4ºK5ºK1ºK6ºK7ºK8ºK9ºK10ºK11ºK12ºK13ºK14ºK50ºK15ºK16ºK17ºK18ºK19ºK20ºK

Ec

Ev

Ea

Acceptor


Extrinsic Material

h+

Al

e- Sb

Si


Carriers Concentrations

• In calculating semiconductor electrical properties and analyzingdevice behavior, it is often necessary to know the number ofcharge carriers per cm3 in the material. The majority carrierconcentration is usually obvious in heavily doped material, sinceone majority carrier is obtained for each impurity atom (for thestandard doping impurities).

The concentration of minority carriers is not obvious, however, noris the temperature dependence of the carrier concentration.


Fermi-Dirac distribution function

• Electrons in solids obey Fermi-Dirac statistics.

• In the development of this type of statistics:

Indistinguishability of the electrons

Their wave nature

Pauli exclusion principle

must be considered.

• The distribution of electrons over a range of allowedenergy levels at thermal equilibrium is

kTfEE

eEf

)(

1

1)(

k : Boltzmann’s constant

f(E) : Fermi-Dirac distribution function

Ef : Fermi level


2

1

11

1

1

1)(

)(

kTfEfE

eEf f

Ef

f(E)

1

1/2

E

T=0ºKT1>0ºKT2>T1

Effect of temperature on Fermi level


Function f(E) , the Fermi Dirac distribution function , gives the probability that an available energy state at E will be occupied by an electron at absolute temperature.

Put E = EF in f(E) and we get f(EF) = 1 / 2

Thus an energy state at the Fermi level has a probability of ½ of being occupied by an electron.


Effect of temperature on Fermi level

Every available energy state upto EF is filled at 0K.


Ev

Ec

Ef

E

f(E)01/21

≈≈

f(Ec) f(Ec)

[1-f(Ec)]

Intrinsicn-typep-type

f(E) distribution in intrinsic and extrinsic semiconductors


f(E) distribution in intrinsic and extrinsic semiconductors


kTEE FeEf

/1

1)(

On semiconductors there are two charge carriers: electrons and holes

Electrons on solids obey the Fermi-Dirac statistics. In equilibrium, the electrondistribution over the allowed energy level interval obeys

where EF is called Fermi level For T > 0K the probability to have a state with E=EF occupied, is

2

1

11

1

1

1)(

/

kTEEFFFe

Ef


F

F

EEse

EEseEf

1

0)(

A more detailed review of f(E) indicates that at 0K, thedistribution assumes the rectangular form pictured below.

That means that at 0K anyavailable energy state from upto EF is filled with electronsand every states over EF areunoccupied

For T> 0K there’s a finite probability, f(E), that the states over EFare filled (e.g. T=T1) and a corresponding probability, [1-f(E)], thatthe states below EF are unoccupied


Electron and hole concentrations at equilibrium

To know the concentrations of electrons and holes in a semiconductor, we need to know the densities of available states .

e.g. Conc. Of electrons in CB is

Density of energy states (states/cm3) in the energy range dE

cE

o dEENEfn )().(

Probability of occupancy


Schematic band diagram , density of states, Fermi Dirac distribution and the carrier concentrations for intrinsic SCs at thermal equilibrium

Thermal equilibrium = No excitations except thermal energy


no = NC e-(EC

-EF

)/ KT (IN CONDUCTION BAND)

Effective density of states

no = Nc f(Ec)

Probability of occupancy

As EF moves closer to the CB, the electron concentration increases.

In n-type semiconductors


Electron and Hole Concentrations at Equilibrium

CE

dEENEfn )()(0

The concentration of electrons in the conduction band is

• N(E)dE : is the density of states (states . cm-3) in the energy range dE.

• The result of the integration is the same as that obtained if we repres-ent all of the distributed electron states in the conduction band edge EC.

P.Ravindran, PHY02E Semiconductor Physics, 31 January 2013: Carriers in Semiconductor21 December 2015

MEL G631(L1)

BITS, Pilani23

po = NV e-(EF– E

V)/KT (In Valence Band)

Similarly conc. of holes in the valence band is

po = Nv [ 1 – f(Ev) ]

Probability of finding an empty state at Ev.

Hole concentration increases as EF moves closer to the valence band.


Schematic band diagram , density of states, Fermi Dirac distribution and the carrier concentrations for n-type SCs at thermal equilibrium


po = NV e-(EF– E

V)/KT (IN VALENCE BAND)

no = NC e-(EC

-EF

)/ KT(IN CONDUCTION BAND)

Equations are valid , whether the material is intrinsic or doped, provided thermal equilibrium is maintained.

no po= [NC e-(EC

-EF

)/ KT].(NV e-(EF– E

V)/KT )


For an intrinsic material, EF lies at some intrinsic level . Hence

EF = Ei

Thus for an intrinsic material , electron and hole concentrations are

ni = NC e-(EC

-Ei) / KT

pi = NV e-(Ei– E

V) /KT

Find product of ni and pi from here :


Since ni = pi

We Get

ni = (Nc Nv )1/2 e-(Eg / 2KT)

This Is Called Mass-action Law

-Shows that intrinsic carrier concentration varies with

temperature



EC

EV

Ef

E

Holes

Electrons

Intrinsicn-typep-type

N(E)[1-f(E)]

N(E)f(E)



kTFECE

kTFECE

ee

Ef C

)(

)(

1

1)(

kTFECE

eNn C

)(

0

23

) 2

(22

*

h

kTmN n

C



23

) 2

(22

*

h

kTmN

p

V

)](1[0 VV EfNp

kTVEFE

kTFEVE

ee

Ef V

)(

)(

1

11)(1

kTVEFE

eNp V

)(

0


Calculation of carrier concentration

• The results is:

• If there is no doping: n = p = ni

– it is called intrinsic material

kT

EETconstn Fcexp2/3

kT

EETconstp vFexp2/3

vFFc EEEE

2

vcF

EEE

= Ei

EF: Fermi-level


.:: CALCULATION• Consider 1 cm3 of Silicon. How many atoms does this contain ?

• Solution:

The atomic mass of silicon is 28.1 g which contains Avagadro’s number of atoms.

Avagadro’s number N is 6.02 x 1023 atoms/mol .

The density of silicon: 2.3 x 103 kg/m3

so 1 cm3 of silicon weighs 2.3 gram and so contains

This means that in a piece of silicon just one cubic centimeter in

volume , each electron energy-level has split up into 4.93 x 1022

smaller levels !

23226.02 10

2.3 4.93 1028.1

atoms

Variation of Energy Bands with Alloy Compositionfolk.uio.no/ravi/cutn/semiphy/7.l8_carrier_const.pdfThe atomic mass of silicon is 28.1 g which contains Avagadro’s number of atoms.

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