EE143 S06
Semiconductor Tutorial 2
-Electron Energy Band
- Fermi Level
-Electrostatics of device charges
Professor N Cheung, U.C. Berkeley
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EE143 S06
Semiconductor Tutorial 2
The Simplified Electron Energy Band Diagram
Professor N Cheung, U.C. Berkeley
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EE143 S06
Semiconductor Tutorial 2
Energy Band Diagram with E-field
Electron
Electron
Energy
Energy
E-field
+
2
1
Electric potential
φ(2) < φ(1)
+
EC
2
EV
x
E-field
-
1
EC
EV
Electric potential
φ(2) > φ(1)
x
Electron concentration n
qφ ( 2 ) / kT
n( 2) e
q[ φ ( 2 ) − φ ( 1 )] / kT
= qφ( 1) / kT = e
n(1) e
Professor N Cheung, U.C. Berkeley
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EE143 S06
Semiconductor Tutorial 2
The Fermi-Dirac Distribution (Fermi Function)
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
f(E)
T=0K
0.5
E -Ef
Professor N Cheung, U.C. Berkeley
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EE143 S06
Semiconductor Tutorial 2
Properties of the Fermi-Dirac Distribution
Probability
of electron state
at energy E
will be occupied
(1) f(E) ≅ exp [- (E- Ef) / kT] for (E- Ef) > 3kT
•This approximation is called Boltzmann approximation
Note:
At 300K,
kT= 0.026eV
(2) Probability of available states at energy E NOT being occupied
1- f(E) = 1/ [ 1+ exp (Ef -E) / kT]
Professor N Cheung, U.C. Berkeley
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EE143 S06
Semiconductor Tutorial 2
How to find Ef when n(or p) is known
Ec
q|ΦF|
Ef
(n-type)
Ei
Ev
Ef
(p-type)
n = ni exp [(Ef - Ei)/kT]
Let qΦF ≡ Ef - Ei
∴n = ni exp [qΦF /kT]
Professor N Cheung, U.C. Berkeley
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EE143 S06
Semiconductor Tutorial 2
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
Professor N Cheung, U.C. Berkeley
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EE143 S06
Semiconductor Tutorial 2
The Fermi Energy at thermal equilibrium
At thermal equilibrium ( i.e., no external perturbation),
The Fermi Energy must be constant for all positions
Electron energy
Material A Material B Material C Material D
EF
Position x
Professor N Cheung, U.C. Berkeley
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EE143 S06
Semiconductor Tutorial 2
Electron Transfer during contact formation
System 1
Before
contact
formation
EF1
EF
System 2
Net
negative
charge
E
Professor N Cheung, U.C. Berkeley
EF2
e
EF2
+ Net
positive + charge + -
System 2
System 1
e
System 1
After
contact
formation
System 2
EF1
System 2
System 1
-
EF
+
+
+
E
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EE143 S06
Semiconductor Tutorial 2
Applied Bias and Fermi Level
Fermi level of the side which has a relatively higher electric potential will have a
relatively 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.
q Va
E f1
Side 2
Side 1
+
Va > 0
-
Ef 2
E f1
q| Va |
Ef
2
Side 2
Side 1
+
Va < 0
-
Potential difference across depletion region
= Vbi - Va
Professor N Cheung, U.C. Berkeley
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EE143 S06
Semiconductor Tutorial 2
PN junctions
Thermal Equilibrium
NA
- and
E-field
p
ρ(x) is 0
ND+ and n
Depletion region
Quasi-neutral
region
--
++
--
++
p-Si
NA- only
ρ(x) is -
ρ(x) is 0
Quasi-neutral
region
n-Si
ND+ only
ρ(x) is +
Complete Depletion Approximation used for charges inside depletion region
r(x) ≈ ND+(x) – NA-(x)
http://jas.eng.buffalo.edu/education/pn/pnformation2/pnformation2.html
Professor N Cheung, U.C. Berkeley
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EE143 S06
Semiconductor Tutorial 2
Electrostatics of Device Charges
1) Summation of all charges = 0 ρ2 • xd2 = ρ1 • xd1
2) E-field =0 outside depletion regions
ρ (x)
p-type
Semiconductor
n-type
Semiconductor
ρ2
- xd1
- ρ1
E=0
Professor N Cheung, U.C. Berkeley
x
xd2
x=0
E ≠0
E=0
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EE143 S06
Semiconductor Tutorial 2
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
Professor N Cheung, U.C. Berkeley
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EE143 S06
Semiconductor Tutorial 2
Example Analysis : n+/ p-Si junction
ρ (x)
+Q'
Emax =qNaxd/εs
p-Si
n+ Si
x
x
Depletion
region
Depletion region
-qNa
is very thin and is
approximated as
E (x)
d
x=0
3) ⇒
Slope = qNa/εs
xd
2) ⇒ E = 0
a thin sheet charge
1) ⇒ Q’ = qNaxd
Professor N Cheung, U.C. Berkeley
4) ⇒ Area under E-field curve
= voltage across depletion region
= qNaxd2/2εs
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EE143 S06
Semiconductor Tutorial 2
Superposition Principle
ρ(x)
If ρ1(x) ⇒ E1(x) and V1(x)
ρ2(x) ⇒ E2(x) and V2(x)
then
ρ1(x) + ρ2(x) ⇒ E1(x) + E2(x) and
V1(x) + V2(x)
+qND
-xp
x
-qN- A
+xn
x=0
ρ2(x)
ρ1(x)
+qND
Q=+qNAxp
-xp
x
-qNA
x
+xn
Q=-qNAxp
x=0
Professor N Cheung, U.C. Berkeley
+
x=0
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EE143 S06
Semiconductor Tutorial 2
ρ1(x)
-xp
ρ2(x)
+qND
Q=+qNAxp
x
x
+xn
-qNA
Q=-qNAxp
x=0
x=0
E1(x)
E2(x)
-xp
+xn
x
Slope =
- qNA/εs
x
-
-
Professor N Cheung, U.C. Berkeley
+
x=0
Slope =
+ qND/εs
x=0
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EE143 S06
Semiconductor Tutorial 2
Sketch of E(x)
E(x) = E1(x)+ E2(x)
-xp
x=0
Slope = - qNA/εs
+xn
x
Slope = + qND/εs
-
Emax = - qNA xp /εs
= - qND xn /εs
Professor N Cheung, U.C. Berkeley
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EE143 S06
Semiconductor Tutorial 2
Depletion Mode :Charge and Electric Field Distributions
by Superposition Principle of Electrostatics
ρ (x)
Q'
Metal
ρ (x)
Q'
Oxide
Semiconductor
x=xo + x
d x
Metal
ρ (x)
Q'
Oxide
x=0
E(x)
Metal
Oxide Semiconductor
Metal
E(x)
Oxide
x=xo
Professor N Cheung, U.C. Berkeley
x=0
x=xo
Semiconductor
=
Metal
x
x=x + x
o d
x=xo + x
d
x=0
−ρ
- Q'
x=xo
x
Semiconductor
x=xo + x
d x
Oxide
x
−ρ
x=0
Metal
Semiconductor
x=0
x=x
o
E(x)
Oxide
x=xo
Semiconductor
+
x
x=x o + x
x=0
x=x
d
o
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EE143 S06
Semiconductor Tutorial 2
Why xdmax ~ constant beyond onset of strong inversion ?
VG = VFB + VOX + VSi
Higher than VT
Picks up all
the changes
in VG
Justification:
If surface electron density
changes by ∆n
Ei
Ef
δ
δ ∝ ln (∆ n
∆n ∝ e
Professor N Cheung, U.C. Berkeley
Approximation assumes
VSi does not change much
δ
∆VOX
)
∆n
∝
COX
but the change of VSi changes
only by kT/q [ ln (∆n)] – small!
kT
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EE143 S06
Semiconductor Tutorial 2
N (surface) = n (bulk) • exp [ qVSi/kT]
p-Si
Na=1016/cm3
n-bulk = 2.1 • 104/cm3
n-surface
qVSi
Ei
0.35eV
Ef
Onset of strong inversion
(at VT)
Professor N Cheung, U.C. Berkeley
VSi
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
n-surface
2.10E+04
9.84E+05
4.61E+07
2.16E+09
1.01E+11
4.73E+12
2.21E+14
1.04E+16
4.85E+17
2.27E+19
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