electron and hole equilibrium concentrations 18 and 20 february 2015 chapter 4 topics-two burning...
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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?
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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
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Also Refer To Appendix B
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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
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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
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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
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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
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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
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Fermi-Energy Levels In Extrinsic Semiconductors
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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
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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:
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2020
Variation of Fermi-Energy with Doping Concentration
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2121
Variation of Fermi-Energy with Temperature