semiconductor tutorial 2
TRANSCRIPT
8/6/2019 Semiconductor Tutorial 2
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1Professor N Cheung, U.C. Berkeley
Semiconductor Tutorial 2EE143 S06
-Electron Energy Band
- Fermi Level
-Electrostatics of device charges
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2Professor N Cheung, U.C. Berkeley
Semiconductor Tutorial 2EE143 S06
The Simplified Electron Energy Band Diagram
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3Professor N Cheung, U.C. Berkeley
Semiconductor Tutorial 2EE143 S06
x
ElectronEnergy
E-field +-
EC
EV
21
x
ElectronEnergy
E-field + -
EC
EV
21
Electric potentialφ(2) < φ(1)
Electric potentialφ(2) > φ(1)
Energy Band Diagram with E -field
Electron concentration n
kT/)]1()2([q
kT/)1(q
kT/)2(q
ee
e
)1(n
)2(n φ
φ
φ
=
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4Professor N Cheung, U.C. Berkeley
Semiconductor Tutorial 2EE143 S06
Probability of available states at energy E being occupied
f(E) = 1/ [ 1+ exp (E- Ef ) / kT]where Ef is the Fermi energy and k = Boltzmann constant=8.617 ∗ 10-5 eV/K
The Fermi-Dirac Distribution (Fermi Function)
T=0K
0.5
E -Ef
f ( E )
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5Professor N Cheung, U.C. Berkeley
Semiconductor Tutorial 2EE143 S06
(2) Probability of available states at energy E NOT being occupied
1- f(E) = 1/ [ 1+ exp (Ef -E) / kT]
Properties of the Fermi-Dirac Distribution
(1) f(E) ≅ exp [- (E- Ef ) / kT] for (E- Ef ) > 3kT Note:
At 300K,
kT= 0.026eV•This approximation is called Boltzmann approximation
Probability
of electron state
at energy Ewill be occupied
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Semiconductor Tutorial 2EE143 S06
Ec
Ev
Ei
Ef
(n-type)
Ef
(p-type)
q|ΦF|
Let qΦF ≡ Ef - Ei
n = ni exp [(Ef - Ei)/kT]
∴n = ni exp [qΦF /kT]
How to find Ef when n(or p) is known
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Semiconductor Tutorial 2EE143 S06
Dependence of Fermi Level with Doping Concentration
Ei ≡ (EC+EV)/2 Middle of energy gap
When Si is undoped, Ef = Ei ; also n =p = ni
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Semiconductor Tutorial 2EE143 S06
At thermal equilibrium ( i.e., no external perturbation),
The Fermi Energy must be constant for all positions
The Fermi Energy at thermal equilibrium
Material A Material B Material C Material D
EF
Position x
Electron energy
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Semiconductor Tutorial 2EE143 S06
After contact
formation
Before
contactformation
Electron Transfer during contact formation
E
System 1 System 2
EF1
EF2e
System 1 System 2
EF
-
--
+
++
E
System 1 System 2
EF1
EF2
e
System 1 System 2
EF
-
-
-
+
+
+
Net
negativecharge
Net
positive
charge
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Semiconductor Tutorial 2EE143 S06
Fermi level of the side which has a relatively higher electric potential will have arelatively lower electron energy ( Potential Energy = -q • electric potential.)
Only difference of the E 's at both sides are important, not the absolute position
of the Fermi levels.
Side 2Side 1
Ef1
f 2E
Va > 0
qVaVa
Side 2Side 1
Ef1
f 2E
Va < 0
q| |
+ - -+
Potential difference across depletion region
= Vbi - Va
Applied Bias and Fermi Level
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Semiconductor Tutorial 2EE143 S06
PN junctions
Complete Depletion Approximation used for charges inside depletion region
r(x) ≈ ND+(x) – NA
-(x)
n-Sip-Si
Depletion region
ND+ onlyNA
- only
ρ(x) is +ρ(x) is -
ND+ and n
ρ(x) is 0
NA- and p
ρ(x) is 0
E -field
++
++
- -
- -Quasi-neutral
region
Quasi-neutral
region
n-Sip-Si
Depletion region
ND+ onlyNA
- only
ρ(x) is +ρ(x) is -
ND+ and n
ρ(x) is 0
NA- and p
ρ(x) is 0
E -field
++
++
- -
- -Quasi-neutral
region
Quasi-neutral
region
http://jas.eng.buffalo.edu/education/pn/pnformation2/pnformation2.html
Thermal Equilibrium
S i d T i l 2EE143 S06
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Semiconductor Tutorial 2EE143 S06
1) Summation of all charges = 0
Electrostatics of Device Charges
ρ1
Semiconducto(x)
ρ2
xd1xd2
Semiconductor
-
-
x=0
p-typen-type
2)E
-field =0 outside depletion regions
ρ2 • xd2 = ρ1 • xd1
E = 0E = 0E ≠0
S i d t T t i l 2EE143 S06
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Semiconductor Tutorial 2EE143 S06
3) Relationship between E -field and charge density ρ(x)
d [ε E (x)] /dx = ρ(x) “Gauss Law”
4) Relationship between E -field and potential φ
E (x) = - dφ(x)/dx
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Semiconductor Tutorial 2EE143 S06
Example Analysis : n+/ p-Si junction
x
-qNa
xd
x=0
+Q'
n+ Sip-Si
ρ (x)
E (x)
xd
1) ⇒ Q’ = qNaxd
3) ⇒
Slope = qNa/εs
4) ⇒ Area under E-field curve
= voltage across depletion region
= qNaxd2/2εs
Emax =qNaxd/εs
Depletion
region
2) ⇒ E = 0
Depletion regionis very thin and is
approximated as
a thin sheet charge
Semiconductor Tutorial 2EE143 S06
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Semiconductor Tutorial 2EE143 S06
x
-
x=0
-qNA
+qND
ρ(x)
-xp
+xn
x=0
x
ρ1(x)
-xp
-qNA
Q=+qNAxp
x
x=0
ρ2(x)
+qND
+xn
Q=-qNAxp
Superposition Principle
If ρ1(x) ⇒ E1(x) and V1(x)ρ2(x) ⇒ E2(x) and V2(x)
then
ρ1(x) + ρ2(x) ⇒ E1(x) + E2(x) andV1(x) + V2(x)
+
Semiconductor Tutorial 2EE143 S06
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Semiconductor Tutorial 2EE143 S06
x
-
x=0
Slope =
- qNA /εs
E1(x)
-xp
x=0
x
ρ1(x)
-xp
-qNA
Q=+qNAxp
x
x=0
ρ2(x)
+qND
+xn
Q=-qNAxp
+ x
-
x=0
+xn
E2(x)
Slope =
+ qND /εs
Semiconductor Tutorial 2EE143 S06
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Semiconductor Tutorial 2EE143 S06
x
-
x=0
Slope = - qNA /εs
E(x) = E1(x)+ E2(x)
-xp +xn
Slope = + qND /εs
Emax = - qNA xp /εs
= - qND xn /εs
Sketch of E(x)
Semiconductor Tutorial 2EE143 S06
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Semiconductor Tutorial 2EE143 S06
x
x=0
Q'
Metal
ρ(x)
Oxide
x=xo
x
−ρ
xd
x=0
Q'
Metal Semiconductor
ρ(x)
Oxide
x=xo
x=xo+
Q'-
x
−ρ
xd
x=0
Q'
Metal Semiconducto
ρ(x)
Oxide
x=xo
x=xo+
Semiconductor
Depletion Mode :Charge and Electric Field Distributions
by Superposition Principle of Electrostatics
x
xd
x=0
Metal Semiconductor
E(x)Oxide
x=xo
x=xo+
x
xd
x=0
Metal Semiconductor
E(x)
Oxide
x=xo
x=xo+ x
xd
x=0
Metal Semiconductor
E(x)
Oxide
x=xo
x=xo+
=
+
Semiconductor Tutorial 2EE143 S06
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Semiconductor Tutorial 2EE143 S06
Approximation assumes
VSi does not change much
OX
OX
C
nV
∆∝∆
SiOX FBG V V V V ++=
Picks up all
the changesin VG
Justification:
If surface electron density
changes by ∆nEi
Ef
( )
kTen
nln
δ
∝
∆δ but the change of VSi changes
only by kT/q [ ln (∆n)] – small!
Why xdmax ~ constant beyond onset of strong inversion ?
Higher than VT
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Semiconductor Tutorial 2EE143 S06
VSi n-surface
0 2.10E+040.1 9.84E+05
0.2 4.61E+07
0.3 2.16E+09
0.4 1.01E+110.5 4.73E+12
0.6 2.21E+14
0.7 1.04E+160.8 4.85E+17
0.9 2.27E+19
n-surface
p-Si
Na=1016/cm3
Ef
Ei
0.35eV
n-bulk = 2.1 • 104/cm3
qVSi
Onset of strong inversion
(at VT)
N (surface) = n (bulk) • exp [ qVSi/kT]