nMOSFET Schematic
Four structural masks: Field, Gate, Contact, Metal.
Reverse doping polarities for pMOSFET in N-well.
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nMOSFET Schematic - Solid State Devices | EE 203, Study notes of Solid State Physics
Material Type: Notes; Class: SOLID-STATE DEVICES; Subject: Electrical Engineering; University: University of California-Riverside; Term: Unknown 1989;
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nMOSFET Schematic
Four structural masks: Field, Gate, Contact, Metal.
Reverse doping polarities for pMOSFET in N-well.
nMOSFET Schematic
x
y
p-type substrate
+n source
+n drain
0
L
depletionregion
inversionchannel
W
polysilicon
gate
gateoxide
z
V
g
V
ds
-V
bs
Source terminal: Ground potential.
Gate voltage:
V
g
Drain voltage:
V
ds
Substrate bias voltage:
V
bs
( x
, y
): Band
bending at any point( x
, y
V
( y
): Quasi-Fermi
potential along thechannel. ¾
V
( y
V
( y
L
V
ds
Gradual Channel Approximation
Assumes that vertical field is stronger than lateral field in thechannel region, thus 2-D Poisson’s eq. can be solved in termsof 1-D vertical slices.Current density eq. (both drift and diffusion):Integrate in
x
- and
z
-directions,
where
is the inversion charge/area.
Current continuity requires
I
ds
independent of
y
, integration with
respect to
y
from 0 to
L
yields
J
x y
q
n x y
dV
y
dy
n^
n
)^
)^
μ
I^
y
W
dV dy
Q
y
W
dV dy
Q V
ds
eff
i^
eff
i
(^
)^
(^
)^
(^
Q
y
q
n x y dx
i
x^ i
(^
)^
∫^0
(^
I^
W L
Q V
dV
ds
eff
i
V^
ds
μ
(^
0
Pao-Sah’s Double Integral
Change variable from (
x
, y
) to (
, V
Substituting into the current expression,where
( s
V
) is solved by the gate voltage eq. for a
vertical slice of the MOSFET:
n x y
n
V
n N
e i a
q^
V^
kT
)^
(^
,^
)^
(^
)/
−
ψ
ψ
−
−
−
s B
B s
d V e N n q d
dx d
V
n
q
V
Q
kT V
q
a
i
i
ψ ψ
ψ
ψ ψ
ψ
ψ
ψ
ψ
ψ
) , (
) / ( ) , ( ) (
/)
(
2
E
−
ds
s B
V^
kT V
q
a
i
eff
ds
dV
d
V
e
N
n
W L
q
I^
0
/)
(
2
ψ ψ
ψ
E
V
V
Q C
V
kTN C
q kT
n N
e
g^
fb
s
s ox
fb
s
si^
a
ox
s^
i a
q^
V^
kT
s
=
−
=
−
ψ
ψ
ε
ψ
ψ
2
2 2
1 2
(^
) /
/
Linear Region I-V Characteristics
0
1
2
3
1E+0 1E-2 1E-4 1E-6 1E-8 1E-
0.8 0.6 0.4 0.2 0
Gate Voltage,
(V)
) (arbitrary scale) Log(
(arbitrary scale)dsI Linear
dsI
V^
on
≈V
t
V
g
For
V
ds
V
g
where
is the MOSFET
threshold voltage
V
V
qNC
t^
fb^
B
si^
a^
B
ox
=
2
4
ψ
ε^
ψ
I^
C
W L
V
V
qNC^
V
C
W L
V
V V
ds
eff
ox
g^
fb
B
si^
a^
B
ox
ds
eff
ox
g^
t^
ds
=
−
−
−
^
=
−
μ
ψ
ε
ψ
μ
2
4
(^
)
Saturation Region I-V Characteristics
Drain Voltage
Drain Current
V^ g
1
V^ g
2
V^ g
3
V^ g
4
(V
dsat
,^ I
dsat
)
Keeping the 2nd order terms in
V
ds
where
is the body-effect coefficient.
when V
ds
V
dsat
V
g^
V
)/ t
m
Typically,
m
I^
C
W L
V
V V
m
V
ds
eff
ox
g^
t^
ds
ds
=
−
−
^
μ
(^
)^
2
2
m
qN
C
C C
t W
si^
a^
B
ox
dm ox
ox dm
=
=
=
1
4
1
1
3
ε
ψ /
I^
I^
C
W L
V
V m
ds
dsat
eff
ox
g^
t
=
=
−
μ
(^
(^2) )
2
Pinch-off from Potential Point of View 0
0 Source
Drain
y
L
V(
y)
V
g^
−V
t m
|Q
|/i mC
ox
V^ ds
V^ ds
V^
(y
)
L^ ′
V
y
V
V m
V
V m
y L
V
V m
V
y L
V
g^
t^
g^
t^
g^
t
ds
ds
(^
)^
=
−
−
−
−
−
^
2
2
2
At the pinch-off point, dV
/ dy
Gradual channel approximation breaksdown.Current is injected intothe bulk depletionregion.
Beyond Pinch-off
Subthreshold Currents
(^2) / 1
/)
(
2 2
2
^
=
=
−
−^
kT V
q
i a
s
a
si
s si
s^
s
e
n N
q kT
kTN
Q
ψ
ψ
ε
ε
E
Power series expansion: 1st term
Q
, 2nd term d
Q
, i
−
=
^
^
−
Q
qN
kT q
n N
e
i
si^
a s
i a
q^
V^
kT
s
ε
ψ
ψ
2
2
(^
)/ (^
)
I^
W L
qN
kT q
n N
e
e
ds
eff
si
a s
i a
q^
kT
qV
kT
s^
ds
^
−
ψ
2
2
/^
/
(^
)
I^
C
W L
m
kT q
e
e
ds
eff
ox
q V
V^
mkT
qV
kT
g^
t^
ds
^
^
−^
−
μ
(^
)^
(^
)/
/
2
S
d
I
dV
mkT
q
kT q
C C
ds g
dm ox
=
^
=
=
−
(log
)^
.^
.
1
2 3
2 3
1
Inverse subthreshold slope:
⇒ or,
Body Effect: Dependence of Threshold
Voltage on Substrate Bias
If
V
bs
[^
]
I^
C
W L
V
V
V
V
qN C
V
V
V
ds
eff
ox
g^
fb
B
ds
ds
si^
a
ox
B^
bs
ds
B^
bs
=
−
−
−
^
^
−
−
μ
ψ
ε^
ψ
ψ
2
2
2
2 3
2
2
3 2
3 2
(^
)^
(^
)
/^
/
0
2
4
6
8
10
0.8 0.
1 1.8 1.6 1.4 1.
Substrate Bias Voltage,
(V)
(V) Threshold Voltage,
V
bs
tV
N
=3a
×^10
15
cm
−^3
t^ ox
=200 Å V
=0fb
N
=10a
16
cm
−^3
V
V
qN
V
C
t^
fb^
B
si^
a^
B^
bs
ox
=
2
2
2
ψ
ε^
ψ (^
)
dV dV
qN
V
C
t bs
si^
a^
B^
bs
ox
=
ε
ψ
/^
(^
)
2 2 ⇓ ⇓
MOSFET Channel Mobility
μ
μ
eff
n
x
x
n x dx n x dx i
i
=
∫
∫
(^
)
(
)
0
0
It was empirically found that when
eff
is plotted against an
effective normal field
E
eff
, there exists a “universal
relationship” independent of the substrate bias, dopingconcentration, and gate oxide thickness (Sabnis andClemens, 1979).Here
=
i
d
si
eff
Q
Q
1 2
(^1) ε
E
Since
and
Q
| ≈ i
C
ox
( V
g^
V
), t
Q
qN
C
V
V
d^
si^
a^
B^
ox
t^
fb
B
=
=
−
−
4
2
ε
ψ
ψ
(^
) ox
t
g
ox
B
fb
t
t
V
V
t V
V
6
3
2
−
−
−
=
ψ
eff E
ox
t
g
ox t
t
V
V
t
V
6
3
(^2). 0
−
= eff E
For n
poly gated nMOSFET,
⇒
N-channel MOSFET Mobility
Low field region (low electron density): Limited byimpurity or Coulombscattering (screened at highelectron densities).
Intermediate field region: Limited by phononscattering,
High field region (> 1 MV/cm): Limited by surfaceroughness scattering (lesstemp. dependence).
(^3) / 1
32500
−
×
≈
E
eff
P-channel MOSFET Mobility
i
d
si
eff
Q
Q
(^1) ε
E
In general, pMOSFET mobility does not exhibit as“universal” behavior as nMOSFET.
Electron and Hole Mobilities vs. Field
1
2
3
50 30 500 300200 100 1000
(MV/cm)
(cm /V-s)
Electron
Hole
0.1 μm CMOS
1 μm CMOS
E
eff
2
μ
eff
E
eff
-0.
E
eff
-0.
E
eff
E
eff
t^
=35 Å
t^
=70 Å
ox ox