Carrier Concentrations

Presented by Chanam Lee

August 22nd

Carrier Concentration

� Carrier Properties

� State and Carrier Distributions

� Equilibrium Carrier Concentration

� Carrier Concentration for the Quantum Well Devices

Carrier Properties

� Carrier Movement in Free Space

� Carrier Movement Within the Crystal

� Intrinsic Carrier Concentration

� Extrinsic n-Type Semiconductor

� Extrinsic p-Type Semiconductor

Electronic Materials

� Two-dimensional representation of an Individual Si atom.

Represents each valence electronAsGeGa

PSiAl

CB

VIVIIIValence

Elemental semiconductors

Semiconductors

� When Si (or Ge & GaAs) atoms contact other Si atoms, they form a tetrahederal

� 2D representation of lattice structure:

Electronic Materials

Two dimensions Three dimensions

Carrier Movement in Free Space

Newton’s second law

dtdv

mqEF 0=−=

Carrier Movement Within the Crystal

� Electrons moving inside a semiconductor crystal will collide with semiconductor atoms ==> behaves as a “wave” due to the quantum mechanical effects

� The electron “wavelength” is perturbed by the crystals periodic potential

Carrier Movement Within the Crystal

(electron effective mass) (hole effective mass)

Density of States Effective Masses at 300 K

0.520.066GaAs

0.360.55Ge

0.811.18Si

Material

dtdv

mqEF n*=−=

*nm

dtdv

mqEF p*=−=

*pm

0* / mmp0

* / mmn

Intrinsic Carrier Concentration

� Contains an insignificant concentration of impurity atoms

� Under the equilibrium conditions, for every electron is created, a hole is created also

n = p = ni

� As temperature is increased, the number of broken bonds (carriers) increases

� As the temperature is decreased, electrons do not receive enough energy to break a bond and remain in the valence band.

Extrinsic n-Type Semiconductor

� Donors (Group V): The 5th in a five valence electrons is readily freed to wander about the lattice at room temperature

� There is no room in the valence band so the extra electron becomes a carrier in the conduction band

� Does NOT increase the number of hole concentration

Extrinsic p-Type Semiconductor

� Acceptors (Group III) : three valence electrons readily accept an electron from a nearby Si-Si bond

� Completing its own bonding creates a hole that can wander about the lattice

� Does NOT increase the number of electron concentration

State and Carrier Distribution

� How the allowed energy states are distributed in energy

� How many allowable states were to be found at any given energy in the conduction and valence bands?

� Essential component in determining carrier distributions and concentration

� Density of States

� Fermi Function

� Dopant States

Density of States:

� represents the number of conduction band states lying in the energy range between E and E + dE

� represents the number of valence band states lying in the energy range between E and E + dE

,)(2

)( 32

**

�πcnn

c

EEmmEg

−= cEE ≥

,)(2

)( 32

**

�πEEmm

Egvpp

v

−= vEE ≤

0)()( == vvcc EgEg

dEEgc )(

dEEgv )(

3/ cm

3/ cm

eVcm

StatesofNumber /3 ��

���

�

Fermi Function (I)

� How many of the states at the energy E will be filled with an electron

� f(E), under equilibrium conditions, the probability that an available state at an energy E will be occupied by an electron

� 1- f(E), under equilibrium conditions, the probability that an available state at an energy E will NOT be occupied by an electron

kTEE FeEf /)(1

1)( −+

=

Fermi Function (II)� If

� If

� If

� At T=0K (above), No occupation of states above EF and complete occupation of states below EF

� At T>0K (below), occupation probability is reduced with increasing energy f(E=EF) = 1/2 regardless of temperature.

2/1)(, == FF EfEE

,3 TkEE BF +≥]/)exp[()( TkEEEf BF −≈

,3 TkEE BF −≤]/)exp[(1)( TkEEEf BF−−≈

� At higher temperatures, higher energy states can be occupied, leaving more lower energy states unoccupied (1-f(E))

Fermi Function (III)

Dopant States

n-type: moreelectrons thanHoles

Intrinsic:Equal numberof electronsand holes

p-type: moreholes thanelectrons

Density of States

Occupancy factors

Carrier Distribution

Equilibrium Carrier Concentration

� Formulas for n and p

� Degenerate vs. Non-degenerate Semiconductor

� Alternative Expressions for n and p

� ni and the np Product

� Charge Neutrality Relationship

� Carrier Concentration Calculations

� Determination of EF

Formulas for n and p

�= top

c

E

E c dEEfEgn )()(

� −+−

= top

c F

E

E kTEEcnn dE

e

EEmmn /)(32

**

1)(2

�π

kTEE

andkT

EELetting cF

cc −=−= ηη

0, == ηcEEwhen

∞→TopELet

�∞

−+=

E

E

nn de

kTmmn

c0)(

2/1

32

2/3**

1)(2

ηηπ ηη

�

)(2/1 cF η

2/3

2

*2/3

2

* 22,

22

��

���

=�

��

=

h

kTmN

hkTm

N pv

nc

ππ

( )cc FNn ηπ 2/1

2≡

� −= v

bottom

E

E v dEEfEgp )](1)[(

( )vv FNp ηπ 2/1

2≡

Degenerate vs. Non-degenerate Semiconductor

( ) kTEEc

cFeF /)(2/1 2

−= πη

TkEEv

TkEEc

BFvBcF eNpeNn /)(/)( , −− ==

,3 TkEEIf BcF −< )()(1

1)( c

ce

eEf ηη

ηη−−

− ≅+

=

( ) kTEEv

FveF /)(2/1 2

−= πη,3 TkEEIf BvF +>

Alternative Expressions for n and p

When n=ni, EF = Ei, thenTkEE

vTkEE

ciBivBci eNeNn /)(/)( −− ==

TkEEc

BcFeNn /)(0

−= TkEEv

BFveNp /)(0

−=

TkEEi

TkEEEEi

BiFBcFic enenn /)(/)(0

−−+− ==TkEE

iTkEEEE

iBFiBFvvi enenp /)(/)(

0−−+− ==

200 inpn =

TkEEiv

TkEEic

BviBic enNenN /)(/)( , −− ==

ni and the np Product

TkEvc

TkEEvci

BgBvc eNNeNNn //)(2 −−− ==

TkEEv

TkEEci

BivBci eNeNn /)(/)( −− ==

TkEvci

BgeNNn 2/−=

Charge Neutrality Relationship

� For uniformly doped semiconductor:Charge must be balanced under equilibrium conditions otherwise charge would flow

Thermally generated + assume ionization of dopant addition all dopant sites

0=+−− +−DA qNqNqnqp

Carrier Concentration Calculations

0)()( =−+− nNNp DA

2/1

2

22/1

2

2

22,

22 ��

��

�

+��

�

����

� −+−=��

��

�

+��

�

����

� −+−= iDADA

iADAD n

NNNNpn

NNNNn

2innp =

0)()(2

=−+− nNNn

nDA

i

0)( 22 =−−− iAD nNNnn

Relationship for and +DN −

AN

kTEED

DDkTEE

A

AA DFFA g

NN

gN

N /)(/)( 1,

1 −+

−−

+=

+=

� The degeneracy factors account for the possibility of electronswith different spin, occupying the same energy level (no electron with the same quantum numbers can occupy the same state)

� Most semiconductor gD=2 to account for the spin degeneracy at the donor sites

� gA is 4 due to the above reason combined with the fact thatthere are actually 2 valence bands in most semiconductorsThus, 2 spins x 2 valance bands makes gA=4

Determination of EF (Intrinsic Material)

TkEEv

TkEEc

BivBci eNeN /)(/)( −− =

peNeNn TkEEv

TkEEc

BivBci === −− /)(/)(

2/3

2

*2/3

2

* 22,

22

��

���

=�

��

=

h

kTmN

hkTm

N pv

nc

ππ

���

����

�++=

c

vBvci N

NTkEEE ln

22

��

�

�

��

�

�++= *

*

ln4

32 n

pBvci m

mTkEEE

2/1

*

*

��

�

�

��

�

�=

n

p

c

v

m

m

NN

Determination of EF (Extrinsic Material)

���

����

�=−

i

DiF n

NkTEE ln

���

����

�−=��

�

����

�=−

iiiF n

pkT

nn

kTEE lnln

���

����

�−=−

i

AiF n

NkTEE ln

iDAD nNandNNforor >>>> iADA nNandNNforor >>>>

TkEEi

TkEEi

BFiBiF enpenn /)(/)( , −− ==

Determination of EF (Extrinsic Material)

Fermi level positioning in Si at room temperature as a function of the doping concentration. Solid EF lines were established using Eq. (previous page)

Carrier Concentration for the Quantum Well Devices

� Density of States 3D vs. 2D

� Carrier Concentration – 2D

� Charge Neutrality

Density of States 3D vs. 2D

32

2)(

�πmEm

Eg =

� 3D

Energy Dependent

� 2DEnergy Independent

ZLm

g 32�π

=

Carrier Concentration – 2D

[ ]� −+=i

kTEE

z

n iFCeL

kTmn /)(

32

*

1ln�π

[ ]� −+=i

kTEE

z

p iFVeL

kTmp /)(

32

*

1ln�π

�= top

c

E

E c dEEfEgn )()(

Charge Neutrality

Llhhh NNNPP ++=+ ΧΓ

References

� Robert F. Pierret, Semiconductor Fundamentals (VOLUME I), Addison-Wesley Publishing Company, 1988, chapter 2

� Robert F. Pierret, Advanced Semiconductor Fundamentals (VOLUME VI), Addison-Wesley Publishing Company, 1987, chapter 4

� Ben G. Streetman and Sanjay Banerjee, Solid State Electronic Devices, Prentice Hall, Inc., 2000, chapter 3

� Peter S. Zory, JR., Quantum Well Lasers, Academic Press, 1993, chapter 1, 7

Effective Mass of Holes - 3D

32

**

32

** )(2)(2�� ππ

EEmmEEmm vlhlhvhhhh −+

−=

[ ] 3/22/3*2/3** )()( lhhhp mmm +=

32

** )(2)(

�πEEmm

Eg vppv

−=

2/3*2/3*2/3* )()()( lhhhp mmm +=