EE143 F2010
Microfabrication
controls dopant
concentration
distribution
ND(x) and NA(x)
Lectre 21
Electron Concentration n(x)
Hole Concentration p(x)
Carrier Mobility
Fermi level Ef (x) , Electric Field
Electrical resistivity
PN Diode Characteristics
Sheet Resistance
MOS Capacitor
Professor N Cheung, U.C. Berkeley
MOS Transistor
1
EE143 F2010
Lectre 21
Electron and Hole Concentrations
for homogeneous semiconductor at thermal equilibrium
n: electron concentration (cm-3)
p : hole concentration (cm-3)
ND: donor concentration (cm-3)
NA: acceptor concentration (cm-3)
Assume completely
ionized to form ND+
and NA-
1) Charge neutrality condition:
ND + p = NA + n
2) Law of Mass Action
n p = ni2
:
Note: Carrier concentrations depend on
NET dopant concentration (ND - NA) !
Professor N Cheung, U.C. Berkeley
2
EE143 F2010
Lectre 21
How to find n, p when Na and Nd are known
n- p = Nd - Na (1)
pn = ni2
(2)
(i) If Nd -Na > 10 ni : n  Nd -Na
(ii) If Na - Nd > 10 ni : p  Na- Nd
* Either n or p will dominate for typical doping situations
Professor N Cheung, U.C. Berkeley
3
EE143 F2010
Lectre 21
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
4
EE143 F2010
Properties of the Fermi-Dirac Distribution
Lectre 21
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
5
EE143 F2010
Lectre 21
Fermi Energy (Ei) of Intrinsic Semiconductor
Ec
Eg /2
Ei
Eg /2
Ev
Professor N Cheung, U.C. Berkeley
6
EE143 F2010
Lectre 21
How to find Ef when n(or p) is known
Ec
q|F|
Ei
Ev
Ef
(n-type)
Ef
(p-type)
n = ni exp [(Ef - Ei)/kT]
Let qF  Ef - Ei
Note: Ef is not
very sensitive to
carrier
concentration
When n doubles,
Ef changes by
kT/q(ln 2) =
18mV
n = ni exp [qF /kT]
Professor N Cheung, U.C. Berkeley
7
EE143 F2010
Lectre 21
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
8
EE143 F2010
Lectre 21
Work Function of Materials
METAL
Work function
= q
SEMICONDUCTOR
Eo
Ef
Vacuum
energy level
q
Ef
Eo
EC
EV
qM is determined
qS is determined
by the metal material
by the semiconductor material,
the dopant type,
and doping concentration
Professor N Cheung, U.C. Berkeley
9
EE143 F2010
Lectre 21
Work Function (qM) of MOS Gate Materials
Eo = vacuum energy level
Ef = Fermi level
EC = bottom of conduction band EV = top of conduction
band
q = 4.15eV (electron affinity)
Eo
Eo
Eo
qM
qM
Ef
Al = 4.1 eV
TiSi2 = 4.6 eV
q = 4.15eV
q = 4.15eV
EC
EC
Ef
0.56eV
Ei
Ei
0.56eV
EV
n+ poly-Si
Professor N Cheung, U.C. Berkeley
qM
0.56eV
Ef
0.56eV
EV
p+ poly-Si
10
EE143 F2010
Lectre 21
Work Function of doped Si substrate
F
* Depends on substrate concentration NB
Eo
Eo
qs
Ef
kT  N B 


ln 
q
 ni 
q = 4.15eV
EC
0.56eV
|q F|
Ei
EV
n-type Si
s (volts) =
4.15 +0.56 - |F|
Professor N Cheung, U.C. Berkeley
qs
Ef
0.56eV
q = 4.15eV
EC
0.56eV
Ei
0.56eV
|qF|
EV
p-type Si
s (volts) =
4.15 +0.56 + |F|
11
EE143 F2010
Lectre 21
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
12
EE143 F2010
Lectre 21
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 1
EF
-
System 2
+
+
+
E
13
EE143 F2010
Lectre 21
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
14
PN junctions
EE143 F2010
Lectre 21
Thermal Equilibrium
NA
- and
E-field
p
(x) is 0
ND+ and n
Depletion region
Quasi-neutral
region
--
++
--
++
p-Si
NA- only
(x) is -
(x) is 0
Quasi-neutral
region
n-Si
ND+ only
(x) is +
Complete Depletion Approximation used for charges inside depletion region
(x)  ND+(x) – NA-(x)
http://jas.eng.buffalo.edu/education/pn/pnformation2/pnformation2.html
Professor N Cheung, U.C. Berkeley
15
EE143 F2010
Lectre 21
Energy Band Diagram with E-field
Electron
Electron
Energy
Energy
E-field
+
2
1
Electric potential
(2) < (1)
+
EC
EV
2
x
E-field
-
1
Electric potential
(2) > (1)
EC
EV
x
Electron concentration n
q ( 2 ) / kT
n( 2 ) e
q[  ( 2 )   ( 1 )] / kT
 q(1) / kT  e
n(1) e
Professor N Cheung, U.C. Berkeley
16
EE143 F2010
Lectre 21
Electrostatics of Device Charges
1) Summation of all charges = 0 2  xd2 = 1  xd1
2) E-field =0 outside depletion regions
(x)
p-type
Semiconductor
- xd1
Professor N Cheung, U.C. Berkeley
x
xd2
- 
E=0
n-type
Semiconductor

x=0
E 0
E=0
17
EE143 F2010
Lectre 21
3) Relationship between E-field and charge density (x)
d [ E(x)] /dx = (x) “Gauss Law”
4) Relationship between E-field and potential 
E(x) = - d(x)/dx
Professor N Cheung, U.C. Berkeley
18
EE143 F2010
Lectre 21
Example Analysis : n+/ p-Si junction
(x)
+Q'
p-Si
n+ Si
x
d
x
Depletion
region
Depletion region
-qNa
is very thin and is
approximated as
Emax =qNaxd/s
x=0
E (x)
3) 
Slope = qNa/s
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
= qNaxd2/2s
19
EE143 F2010
Lectre 21
Superposition Principle
(x)
If 1(x)  E1(x) and V1(x)
2(x)  E2(x) and V2(x)
then
1(x) + 2(x)  E1(x) + E2(x) and
V1(x) + V2(x)
+qND
-xp
x
-qN- A
+xn
x=0
2(x)
1(x)
+qND
Q=+qNAxp
-xp
x
-qNA
+xn
x
Q=-qNAxp
x=0
Professor N Cheung, U.C. Berkeley
+
x=0
20
EE143 F2010
Lectre 21
1(x)
-xp
2(x)
+qND
Q=+qNAxp
x
+xn
-qNA
Q=-qNAxp
x=0
x=0
E1(x)
E2(x)
-xp
x
Slope =
- qNA/s
-
Professor N Cheung, U.C. Berkeley
x
x=0
+
+xn
x
-
x=0
Slope =
+ qND/s
21
EE143 F2010
Lectre 21
Sketch of E(x)
E(x) = E1(x)+ E2(x)
-xp
Slope = - qNA/s
x=0
+xn
x
Slope = + qND/s
-
Emax = - qNA xp /s
= - qND xn /s
Professor N Cheung, U.C. Berkeley
22
EE143 F2010
Lectre 21
The MOSFET
Metal-OxideSemiconductor
GATE LENGTH, L
GATE WIDTH, W
Field-Effect
Transistor
Gate
Source
OXIDE
THICKNESS, Tox
Drain
Substrate
Desired characteristics:
• High ON current
• Low OFF current
VT
CURRENT
• No current through the gate
to channel
• Current flowing between the
SOURCE and DRAIN is
controlled by the voltage on
the GATE electrode
|GATE VOLTAGE|
Professor N Cheung, U.C. Berkeley
EE143 F2010
Lectre 21
n-channel MOSFET
G
S
n
D
gate
oxide insulator
p
n
• Without a gate voltage applied, no current can flow
between the source and drain regions (2 pn diodes
back-to -back)
• Above a certain gate-to-source voltage (threshold
voltage VT), a conducting layer of mobile electrons is
formed at the Si surface beneath the oxide. These
electrons can carry current between the source and
drain.
Professor N Cheung, U.C. Berkeley
EE143 F2010
Lectre 21
The MOSFET as a Controlled Resistor
• When VDS is low, ID increases linearly with VDS
Inversion charge density Qi(x) = -Cox[ VGS – VT - V(x) ]
where Cox 
ox
/ tox.
ID
VGS = 2 V
VGS = 1 V > VT
VDS
IDS = 0 if VGS < VT
Professor N Cheung, U.C. Berkeley
EE143 F2010
Lectre 21
Pinch-off and Saturation
• As VDS increases, V(x) along channel increases.
Inversion-layer charge density Qn at the drain end of the channel is
reduced; therefore, ID increases slower than linearly with VDS.
• When VDS reaches VGS  VT, the channel is “pinched off” at the drain
end, and ID saturates (i.e. it does not increase with further increases
in VDS).
+
–
VGS
VDS > VGS - VT
G
D
S
n+
-
Professor N Cheung, U.C. Berkeley
VGS - VT
+
n+
pinch-off
region
I DSAT
W
VGS  VT 2
  nCox
2L
EE143 F2010
Professor N Cheung, U.C. Berkeley
Lectre 21
EE143 F2010
Lectre 21
NMOS versus PMOS
NMOS
Saturation
Cut-off
0 Vto
Triode
vGS
vDS Vto
PMOS
Triode
Saturation
vDS Vto
Professor N Cheung, U.C. Berkeley
Cut-off
Vto
0
vGS
EE143 F2010
Professor N Cheung, U.C. Berkeley
Lectre 21
EE143 F2010
Professor N Cheung, U.C. Berkeley
Lectre 21
EE143 F2010
Lectre 21
IDsat 
Not proportional to
(VGS-Vt)2
Professor N Cheung, U.C. Berkeley