cse 477. vlsi systems design - guc · dr. hassan mostafa ىفطصم نسح .د...
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ELCT 503: Semiconductors German University in Cairo (GUC)
ELCT503
Semiconductors Fall 2014
Lecture 02: Intrinsic Semiconductors (Cont.)
Dr. Hassan Mostafa
حسن مصطفى. د
ELCT 503: Semiconductors German University in Cairo (GUC)
Contents
Semiconductors
Crystal structure
Band diagram
Intrinsic carrier concentration
Influence of temperature
ELCT 503: Semiconductors German University in Cairo (GUC)
Electrons and holes in si
Eg = 1.12 eV
Conduction band
Valence band
electron
hole
Prohibited band
EC
EV
E
ELCT 503: Semiconductors German University in Cairo (GUC)
INTRINSIC CARRIER CONCENTRATION
Intrinsic = no impurities (Pure Silicon)
Fermi distribution function F(E):
The probability that an electron occupies an electronic state with energy E
Probability that a hole occupies an electronic state with energy E = 1-f(E)
1
1)(
(
kTEE F
EF)/
e
EF Fermi level
k Boltzmann’s constant
T temperature in kelvins
ELCT 503: Semiconductors German University in Cairo (GUC)
INTRINSIC CARRIER CONCENTRATION
Electron density n : the number of electrons per unit volume (Units: cm-3)
n(E) = the electron density at energy level E
N(E) = density of states = number of allowed energy states per energy
range per unit volume
Number of electrons in the conduction band:
This looks like you sum the product of each energy level by the
probability of finding electrons in that energy level
CE
f(E)N(E)dEn
ELCT 503: Semiconductors German University in Cairo (GUC)
Density of energy states
2123
32
4 /
c
/
e )E(E)m(h
πN(E)
2123
32
4 /
v
/
h E)(E)m(h
πN(E)
in the conduction band:
in the valence band:
where: h – Planc’s constant, [h] = J·s=kg ·m2 · s-1
me – effective mass of electron
mh – effective mass of hole
ELCT 503: Semiconductors German University in Cairo (GUC)
Density of energy states
There are a large number of allowed states in the conduction band. However, for an intrinsic semiconductor, the probability of an electron occupying one of these states is small. Thus, there will not be many electrons in the conduction band.
There also are a large number of allowed states in the valence band. the probability of an electron occupying one of these states in the valence band is nearly unity. Thus, most of these energy states are occupied by electrons.
Note that there will be only a few unoccupied electron states in the valence band, that is, holes, in the valence band.
ELCT 503: Semiconductors German University in Cairo (GUC)
Carrier concentration
CE
i f(E)N(E)dEnpn
CB
VB
E
N(E)
EG
CB
VB
E
n(E),p(E)
EG
density of states carrier
concentration
CB
VB
E
F(E)
EG
0 1 0.5
EC
EV
EFi
Fermi distribution
function
n(E)
p(E)
ELCT 503: Semiconductors German University in Cairo (GUC)
Electrons in conduction band
Total number of electrons in the conduction band:
C
FC
E
)/kTE(E
CeNf(E)N(E)dEn
Nc – Effective density of states in the conduction band
23
2
212
/
nC
h
kTmπN
23
2
22
/
nC
h
kTmπN
For silicon: For gallium arsenide:
k – Boltzman’s constant, [k] = J· K-1
ELCT 503: Semiconductors German University in Cairo (GUC)
Holes in valence band
Total number of holes in the valence band:
V
VF
E
)/kTE(E
V eNN(E)dEf(E)p 1
NV – Effective density of states in the valence band
23
2
22
/
hV
h
kTmπN
ELCT 503: Semiconductors German University in Cairo (GUC)
Fermi level in intrinsic semiconductor
C
VVCiFi
N
NkTEEEE ln
22
from:
)/kTE(E
VVFeNp
)/kTE(E
CFCeNn
and:
inpn Hence:
For an intrinsic semiconductor (thermal equilibrium), the number of
electrons per unit volume in the conduction band is equal to the
number of holes per unit volume in the valence band, that is, n = p =
ni, where ni is the intrinsic carrier density.
ELCT 503: Semiconductors German University in Cairo (GUC)
Fermi level in intrinsic semiconductor
C
VVCiFi
N
NkTEEEE ln
22
At room temperature, the second term is much smaller than the
bandgap. Hence, the intrinsic Fermi level E, of an intrinsic
semiconductor generally lies very close to the middle of the bandgap.
ELCT 503: Semiconductors German University in Cairo (GUC)
Intrinsic carrier density
/kTE
VCgeNNpn
2
innp
inpn
kT/ENNn gVCi 2exp
ELCT 503: Semiconductors German University in Cairo (GUC)
Influence of temperature on band gap
1.125
1.424
GaAs
Si
K636KeV1073.4eV170.1
21-4
g
T
TE
K204KeV10405.5eV519.1
21-4
g
T
TE
ELCT 503: Semiconductors German University in Cairo (GUC)
Influence of temperature on ni
300K
1000K 250K
1010
GaAs
Si
ELCT 503: Semiconductors German University in Cairo (GUC)
Summary
/kTE
VCgeNNpn
2
innp
222
g
C
g
VVC
iFi
EE
EE
EEEE
V
VF
E
)/kTE(E
V eNN(E)dEf(E)p 1
C
FC
E
)/kTE(E
CeNf(E)N(E)dEn
ELCT 503: Semiconductors German University in Cairo (GUC)
Short Break
ELCT 503: Semiconductors German University in Cairo (GUC)
ELCT503
Semiconductors Fall 2014
Lecture 03: Extrinsic Semiconductors
Dr. Hassan Mostafa
حسن مصطفى. د
ELCT 503: Semiconductors German University in Cairo (GUC)
content
n-type semiconductors
p-type semiconductors
Mass action law & Charge neutrality
Majority and minority carriers
Influence of temperature on carrier concentration
ELCT 503: Semiconductors German University in Cairo (GUC)
n-type semiconductors
ELCT 503: Semiconductors German University in Cairo (GUC)
n-type semiconductors
+5 - -
+4
+4 +4
- -
- -
+4
- - +4
- - +4
- -
+4 - -
+4 - -
- -
- -
- -
- - -
-
-
-
-
-
- -
-
-
- -
CB
VB
x
EG
E
EC
EV
ED
ED donor energy level
Small binding energy to donor atom
ELCT 503: Semiconductors German University in Cairo (GUC)
n-type semiconductors
EC – ED = Ionization energy
ELCT 503: Semiconductors German University in Cairo (GUC)
Silicon and other semiconductors
Period II III IV V VI
2 B C N O
3 Mg Al Si P S
4 Zn Ga Ge As Se
5 Cd In Sn Sb Te
6 Hg Pb Bi
donors: pentavalent elements from group V (P,As,Sb,Bi)
→ release of electrons → n-type semiconductor
The most famous are P (Phosphorus) and As (Arsenic)
ELCT 503: Semiconductors German University in Cairo (GUC)
ionization of donors
complete ionization
of donors:
ionization energy of impurities
DNn
S.M.Sze
ELCT 503: Semiconductors German University in Cairo (GUC)
Again: Intrinsic carrier concentrations
CB
VB
E
N(E)
CB
VB
E
F(E) 0 0.5 1
E
CB
VB
n(E), p(E)
Fermi distribution
function density of states carrier
concentration
EG EG
ELCT 503: Semiconductors German University in Cairo (GUC)
Extrinsic carrier concentrations (n-type)
CB
VB
E
N(E)
CB
VB
E
F(E) 0 0.5 1
E
CB
VB
n(E), p(E)
Fermi distribution
function density of states carrier
concentration
EG EG
EC
CE
f(E)N(E)dEn
ELCT 503: Semiconductors German University in Cairo (GUC)
Ascending fermi level
DCFC NNkTEE ln
DNn
)/kTE(E
CFCeNn
Valence band
Conduction band
EC
EV
Efi = Ei
ED
EFn
EC – ED = Ionization energy
ELCT 503: Semiconductors German University in Cairo (GUC)
Electron concentration in doped semiconductor
)/kTE(ENn FCC exp
)/kTE(Enn iFi exp
in
)/kTE(E)/kTE(EN FiiCC expexp
)/kTEEE(EN FiiCC exp
Valence band
Conduction band
EC
EV
Efi = Ei
ED
EFn
DCFC NNkTEE ln
EC – ED = Ionization energy
ELCT 503: Semiconductors German University in Cairo (GUC)
Short Break
ELCT 503: Semiconductors German University in Cairo (GUC)
p-type semiconductors
+3 -
+4
+4 +4
- -
- -
+4
- - +4
- - +4
- -
+4 - -
+4 - -
- -
- -
- -
- - -
-
-
-
-
-
- -
-
-
- -
CB
VB
E
x
EG
EC
EV EA
EA acceptor energy level
ELCT 503: Semiconductors German University in Cairo (GUC)
p-type semiconductors
EA – EV = Ionization energy
ELCT 503: Semiconductors German University in Cairo (GUC)
Silicon and other semiconductors
Period II III IV V VI
2 B C N O
3 Mg Al Si P S
4 Zn Ga Ge As Se
5 Cd In Sn Sb Te
6 Hg Pb Bi
acceptors: trivalent elements from group III (B, Al, Ga, In)
→capture of electron→hole remains → p-type semiconductor
The most famous is B (Boron)
ELCT 503: Semiconductors German University in Cairo (GUC)
ionization of impurities
ionization energy of impurities
ANp
complete ionization
of acceptors:
S.M.Sze
ELCT 503: Semiconductors German University in Cairo (GUC)
Intrinsic carrier concentrations
Fermi distribution
function density of states carrier
concentration
CB
VB
E
N(E) 0.5
CB
VB
E
EV
0 1
CB
VB
E
EG
n(E), p(E) F(E)
EG
ELCT 503: Semiconductors German University in Cairo (GUC)
Extrinsic carrier concentrations (p-type)
Fermi distribution
function density of states carrier
concentration
CB
VB
E
N(E) 0.5
CB
VB
E
0 1
CB
VB
E
n(E), p(E) F(E)
VE
N(E)dEf(E)p0
1
EG EG
ELCT 503: Semiconductors German University in Cairo (GUC)
Descending fermi level
AVVF NNkTEE ln
)/kTE(E
VVFeNp
ANp
Valence band
Conduction band
EC
EV
Efi = Ei
EA
EFp
EA – EV = Ionization energy
ELCT 503: Semiconductors German University in Cairo (GUC)
Hole concentration in doped semiconductor
)/kTE(ENp VFV exp
)/kTE(Enp Fii exp
in
)/kTEEE(EN ViiFV exp
)/kTE(E)/kTE(EN iFViV expexp
Valence band
Conduction band
EC
EV
Efi = Ei
EA
EFp
EA – EV = Ionization energy
AVVF NNkTEE ln
ELCT 503: Semiconductors German University in Cairo (GUC)
Mass action law
)/kTE(En)/kTE(Enpn iFiFii expexp
2
inpn
This equation holds for both intrinsic and extrinsic semiconductors
ELCT 503: Semiconductors German University in Cairo (GUC)
charge neutrality
DA NpNn
In general both acceptors and donors can be present
positive ions negative ions
charge neutrality = zero net charge density
ELCT 503: Semiconductors German University in Cairo (GUC)
majority and minority carriers
electrons holes
n-doped
(ND > NA)
MAJORITY
carrier
MINORITY
carrier
p-doped
(ND < NA)
MINORITY
carrier
MAJORITY
carrier
ELCT 503: Semiconductors German University in Cairo (GUC)
Majority and minority carrier (n-type)
majority electrons:
nin nnp /2
22
42
1iADADn nNNNNn
DADnAD NNNnNN
minority holes:
2
inn nnp DnAn NpNn from: and
ELCT 503: Semiconductors German University in Cairo (GUC)
Majority and minority carrier (p-type)
majority holes:
pip pnn /2
22
42
1iDADAp nNNNNp
ADApDA NNNpNN
minority electrons:
2
ipp nnp DpAp NpNn from: and
ELCT 503: Semiconductors German University in Cairo (GUC)
Non-degenerate semiconductor
CFCD EENNn
VFVA EENNp
and
DCFC NNkTEE ln
AVVF NNkTEE ln
The approximations used for f(E) can not be used for degenerate semiconductors
(such as heavily doped semiconductors for which NC < n or NV < p and
Fermi levels lie in the conduction band (n-type) and in the valence band (p-type)),
All the above formulas does not work and the integral should be done numerically
ELCT 503: Semiconductors German University in Cairo (GUC)
Influence of temperature on carrier concentration
ELCT 503: Semiconductors German University in Cairo (GUC)
electron temperature vs. temperature
S.M.Sze