lecture 6
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Lecture 6
Carriers in Semiconductors
2
Carriers in Semiconductors
Ec
Ev
Eg0ºK3ºK2ºK4ºK5ºK1ºK6ºK7ºK8ºK9ºK10ºK11ºK12ºK13ºK14ºK300ºK
15ºK16ºK17ºK18ºK19ºK20ºK
Electron Hole PairE H P
3
3-2-1. Electrons and Holes
N
iiVqJ 0)(
k
Ekj-kj
j` j
j
N
ii VqVqJ )()(
0
J jVq) ( jV)q(
• Equilibrium: no external forces such as voltages, electrical fields, magnetic fields, or temperature gradients are acting on the semiconductor
9
3-3. Carriers Concentrations
• In calculating semiconductor electrical pro-perties and analyzing device behavior, it is often necessary to know the number of charge carriers per cm3 in the material. The majority carrier concentration is usually obvious in heavily doped material, since one majority carrier is obtained for each impurity atom (for the standard doping impurities).
The concentration of minority carriers is not obvious, however, nor is the temperature dependence of the carrier concentration.
10
3-3-1. The Fermi Level
• 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 these
statistical arguments is that the distrib-ution of electrons over a range of allowed energy levels at thermal equilibrium is
11
3-3-1. The Fermi Level
kTfEE
eEf )(
1
1)(
k : Boltzmann’s constant
f(E) : Fermi-Dirac distribution function
Ef : Fermi level
12
3-3-1. The Fermi Level
2
1
11
1
1
1)( )(
kTfEfE
eEf f
Ef
f(E)
1
1/2
E
T=0ºKT1>0ºKT2>T1
13
3-3-1. The Fermi Level
Ev
Ec
Ef
E
f(E)01/21
≈≈
f(Ec) f(Ec)
[1-f(Ec)]
Intrinsicn-typep-type
14
3-3-2. 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 (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.
)(0 CC EfNn
15
3-3-2. Electron and Hole Concentrations at Equilibrium
EC
EV
Ef
E
Holes
Electrons
Intrinsicn-typep-type
N(E)[1-f(E)]
N(E)f(E)
16
3-3-2. Electron and Hole Concentrations at Equilibrium
kTFECE
kTFECE
ee
Ef C
)(
)(
1
1)(
kTFECE
eNn C
)(
0
23
) 2
(22
*
h
kTmN nC
17
3-3-2. Electron and Hole Concentrations at Equilibrium
)](1[0 VV EfNp
kTVEFE
kTFEVE
ee
Ef V
)(
)(
1
11)(1
kTVEFE
eNp V
)(
0
23
) 2
(22
*
h
kTmN pV
18
3-3-2. Electron and Hole Concentrations at Equilibrium
kTvEiE
eNp Vi
)(
kTiEcE
eNn Ci
)(
kT
gEkT
vEcE
eNNeNNpn vcvc
)(
00
kTgE
eNNpn vcii
kTgE
eNNn vci2
2
00 inpn
kTFEiE
enp i
)(
0
kTiEFE
enn i
)(
0
19
3-3-2. Electron and Hole Concentrations at Equilibrium
Example 3-4: A Si sample is doped with 1017 As Atom/cm3. What is the equilibrium hole concentra-tion p0 at 300°K? Where is EF relative to Ei?
20
3-3-2. Electron and Hole Concentrations at Equilibrium
3317
20
0
2
0 1025.210
1025.2
cmn
np i
Answer: Since Nd»ni, we can approximate n0=Nd and
kTiEFE
enn i
)(
0
eVn
nkTEE
iiF 407.0
105.1
10ln0259.0ln
10
170
21
3-3-2. Electron and Hole Concentrations at Equilibrium
Answer (Continue) :
Ev
Ec
EF
Ei1.1eV
0.407eV
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