Electron And Hole Equilibrium Concentrations18 and 20 February 2015

Chapter 4 Topics-Two burning questions:• What is the density of states in the conduction and valence bands?• How do you find an exact position of the Fermi Level?

2

Concentration of Electrons at Equilibrium

kT

EE

h

kTmdE

kT

EEEE

h

mn cFe

E

Fc

e

c

exp2

2exp)2(4

23

2

*

3

23*

0

Thermal-equilibrium concentration of electrons (#/cm3) in the conduction band

kT

EE

kT

EE

kT

EEF

F

FFe

eeEf

1

1

1)(

Assume that EF is within the energy bandgap and ,kTEE Fc ,cEE So for energy levels in the conduction band kTEE F there is

(kT ≈ 25.9 meV for T=300K). So for energy levels in the conduction band

cE

Fc dEEfEgn )()(0

Page 87 Text, Equation 3.69

Also Refer To Appendix B

88

Concentration of Electrons at Equilibrium

kT

EE

h

kTmn cFe exp

22

23

2

*

0

23

2

*22

h

kTmN ec

Define effective density of states in the conduction band

kT

EENn cFc exp0

1010

Concentration of Holes at Equilibrium

Thermal-equilibrium concentration of holes (#/m3) in the conduction band

kT

EE

kT

EE

kT

EEEf F

FF

F exp

exp

1

exp1

1)(1

Assume that EF is within the energy bandgap and ,kTEE vF ,vEE So for energy levels in the conduction band ,kTEEF there is

then for energy levels in the conduction band

vE

Fv dEEfEgp )(1)(0

kT

EE

h

kTmdE

kT

EEEE

h

mp Fvh

EF

vhv

exp2

2exp)2(4

23

2

*

3

23*

0

1111

Concentration of Holes at Equilibrium

kT

EE

h

kTmp Fvh exp

22

2

*

0

23

2

*22

h

kTmN hv

Define effective density of states in the valence band

kT

EENp Fvv exp0

1414

Carrier Concentrations in Intrinsic Semiconductors

For an intrinsic semiconductor, the concentration of electrons in the conduction band is equal to the concentration of holes in the valence band

kT

EEN

kT

EENnn cFi

ccF

ci expexp0

kT

EEN

kT

EENnpp Fiv

vFv

vii expexp0

kT

ENN

kT

EENNn g

vccv

vci expexp2

FiE is the Fermi-level for the intrinsic semiconductor, i.e., intrinsic Fermi-level

For a given semiconductor, at constant temperature, ni is constant

16

The Intrinsic Fermi-Level position

kT

EEN

kT

EEN Fiv

vcFi

c expexpii pn

*

*

ln4

3

2ln

2

1

2 e

hvc

c

vvcFi m

mkT

EE

N

NkT

EEE

midgapvc EEE

2

*

*

ln4

3

e

hmidgapFi m

mkTEE

If ,**

eh mm midgapFi EE

Fermi-Energy Levels In Extrinsic Semiconductors

18

Electron and Hole Concentrations in Extrinsic Semiconductor Summary

,expexpexp

)()(expexp0

kT

EEn

kT

EE

kT

EEN

kT

EEEEN

kT

EENn

FiFi

FiFcFic

FiFFicc

cFc

,expexpexp

)()(expexp0

kT

EEp

kT

EE

kT

EEN

kT

EEEEN

kT

EENp

FFii

FFiFivv

FFiFivc

Fvv

200 exp i

gvc n

kT

ENNpn

For both undoped material and doped

material under equilibrium condition

19

Position of Fermi Energy in Extrinsic Semiconductors

i

Fi

v

v

i

Fi

c

cF

n

pkTE

N

pkTE

n

nkTE

N

nkTEE

00

00

lnln

lnln

,expexp0

kT

EEn

kT

EENn FiF

icF

c

,expexp0

kT

EEp

kT

EENp FFi

iFv

v

Position of Fermi-level:

2020

Variation of Fermi-Energy with Doping Concentration

2121

Variation of Fermi-Energy with Temperature