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The temperature is 300K with
V
T
=
kT/q
=26mV, while the semiconductor is silicon with
E
gap
= 1.1eV, and
the intrinsic concentration
n
i
= 10
10
cm
3
.
np = n
i
2
can be used to estimate the minority concentration.
For a semiconductor resistor with length
l
and cross section
A
, the resistance can be estimated as
(
29
p
n
qp
qn
A
l
A
l
R
μ
ρ
+
=
=
, where
n
and
p
are the electron and hole concentrations,
n
= 1,000cm
2
/Vs
and
p
= 400cm
2
/Vs are the electron and hole mobility.
The minority diffusion length as:
n
n
n
D
L
τ
=
, where
D
n=
n
kT/q
is the diffusivity and
n
=
p
=1
s
is the
minority recombination lifetime.
Diode equations:
W
d
= x
n
+ x
p
bi
D
A
si
V
N
N
q
+
=
1
1
2
0
ε
, N
A
x
p
= N
D
x
n
,
=
2
ln
i
D
A
T
bi
n
N
N
V
V
0
=8.85
×
10
14
F/cm,
si
=11.7 is the relative dielectric constant of silicon,
ox
=3.9, q
=1.6
×
10
19
coul is the
elemental charge,
k
the Boltzmann constant, and
T
the temperature.
For a longbase bipolar junction diode,
I
D
= I
0
(exp(qV
D
/kT) – 1)
, with
+
=
D
i
p
p
A
i
n
n
N
n
L
D
N
n
L
D
qA
I
2
2
0
.
The
minority injection level at the edge of the depletion region can be approximated by
p
n
= p
n0
×
exp(
qV
D
/kT
), which is valid for both forward and reverse biases. Notice that the above equation
needs to replace
L
n
and
L
p
with the geometrical length in the shortbase case.
For nMOSFET with the threshold voltage
V
th
, the drain current
I
D
in the linear and saturation regions
above threshold (
V
GS
> V
th
) are: (notice that
C
ox
ch
= k
n
’
and
λ
=1/V
A
)
(
29
(
29
saturation
)
1
(
2
linear
2
2
2
th
GS
Dsat
DS
DS
th
GS
ch
ox
D
th
GS
Dsat
DS
DS
DS
th
GS
ch
ox
D
V
V
V
V
V
λ
V
V
μ
C
L
W
I
V
V
V
V
V
V
V
V
μ
C
L
W
I

=
≥
+

=

=
<


=
For nMOSFET below the threshold voltage,
(
29
T
th
GS
T
DS
V
V
V
κ
V
V
th
D
e
e
I
L
W
I
/
/
)
1
(
2



=
and
ox
si
C
C
m
+
2245
≡
1
1
κ
Smallsignal parameters
R
in
g
m
R
out
BJT forward active (E at ground)
r
π
=
β
/g
m
= V
T
/I
B
I
C
/V
T
r
o
=V
A
/I
C
MOSFET aboveVth saturation (S at ground)
∞
2I
D
/V
OV
r
o
=V
A
/I
D
g
mn
v
GS
NMOSFET large/small
signal model for the
saturation region.
1
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 Spring '07
 SPENCER
 Microelectronics

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