Electron And Hole Equilibrium Concentrations24 February 2014

Return and discuss Quiz 2

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

kTEE

hkTmdE

kTEEEE

hmn cFe

EF

ce

c

exp22exp)2(423

2

*

3

23*

0

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

kTEE

kTEE

kTEEF

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

55

Concentration of Electrons at Equilibrium

kTEE

hkTm

n cFe exp2

223

2

*

0

23

2

*22

hkTm

N ec

Define effective density of states in the conduction band

kTEE

Nn cFc exp0

77

Concentration of Holes at Equilibrium

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

kTEE

kTEE

kTEE

Ef F

FFF 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

kTEE

hkTmdE

kTEEEE

hmp Fvh

EF

vhv exp22exp)2(4

23

2

*

3

23*

0

88

Concentration of Holes at Equilibrium

kTEE

hkTm

p Fvh exp2

22

*

0

23

2

*22

hkTm

N hv

Define effective density of states in the valence band

kTEE

Np Fvv exp0

1010

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

kTEE

NkTEE

Nnn cFic

cFci expexp0

kTEE

NkTEE

Nnpp Fivv

Fvvii expexp0

kTE

NNkTEE

NNn gvc

cvvci 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

12

The Intrinsic Fermi-Level position

kTEE

NkTEE

N Fivv

cFic expexp

ii pn

*

*

ln43

2ln

21

2 e

hvc

c

vvcFi m

mkT

EENN

kTEE

E

midgapvc EEE

2

*

*

ln43

e

hmidgapFi m

mkTEE

If ,**eh mm midgapFi EE

Fermi-Energy Levels In Extrinsic Semiconductors

14

Electron and Hole Concentrations in Extrinsic Semiconductor Summary

,expexpexp

)()(expexp0

kTEEn

kTEE

kTEEN

kTEEEEN

kTEENn

FiFi

FiFcFic

FiFFicc

cFc

,expexpexp

)()(expexp0

kTEEp

kTEE

kTEEN

kTEEEEN

kTEENp

FFii

FFiFivv

FFiFivc

Fvv

200 exp i

gvc n

kTE

NNpn

For both undoped material and doped

material under equilibrium condition

15

Position of Fermi Energy in Extrinsic Semiconductors

iFi

vv

iFi

ccF

np

kTENp

kTE

nn

kTENn

kTEE

00

00

lnln

lnln

,expexp0

kTEE

nkTEE

Nn FiFi

cFc

,expexp0

kTEEp

kTEENp FFi

iFv

v

Position of Fermi-level:

1616

Variation of Fermi-Energy with Doping Concentration

1717

Variation of Fermi-Energy with Temperature