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ECE 440
Homework VIII Solutions
Spring 2008
1.
An abrupt Si pn junction is formed by alloying a uniformly doped ntype silicon bar of
which N
d
= 4x10
16
/cm
3
. During the alloying process, a uniform counter doping of acceptors
of N
a
= 1.5x10
17
/cm
3
is introduced in the region for x<0. For x>0, the doping remains to be
N
d
= 4x10
16
/cm
3
. So, x<0 is the pside and x>0 is the nside.
(a) Calculate the Fermi level positions at 300 K in the p and n regions.
Treat each side separately (as if they were unconnected) to calculate the Fermi level
positions far from the junction.
pside
:
eV
n
N
N
kT
n
p
kT
E
E
i
d
a
i
f
i
409
.
0
10
5
.
1
10
4
10
5
.
1
ln
0259
.
0
ln
ln
10
16
17
0
=
×
×

×
=

=
=

nside
:
eV
n
N
kT
n
n
kT
E
E
i
d
i
i
f
383
.
0
10
5
.
1
10
4
ln
0259
.
0
ln
ln
10
16
0
=
×
×
=
=
=

(b) Draw an equilibrium band diagram for the junction and determine the contact potential
V
o
from the diagram.
Looking at the band diagram we can calculate qV
0
geometrically. The contact potential is
the difference between the Fermi level positions on the n and p sides of the junction.
eV
eV
eV
F
E
E
F
F
F
qV
p
i
i
n
p
n
792
.
383
.
0
409
.
0
)
(
)
(
0
=
+
=

+

=

=
(c) Compare the results of part (b) with V
o
as calculated from Eq. (58).
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View Full DocumenteV
kT
n
N
N
kT
qV
i
d
a
792
.
)
10
5
.
1
(
)
10
4
(
)
10
4
10
5
.
1
(
ln
ln
2
10
16
16
17
2
0
=
×
×
⋅
×

×
=
⋅
=
In fact, equation (58) can easily be derived from the expression in part (b).
(d) Using Eq. (58), calculate and plot V
o
versus temperature ranging from 250 K to 500 K.
Looking at equation (58), there is a linear dependence between contact potential and
temperature. However, the intrinsic carrier concentration, n
i
(T), also has a temperature
dependence (see equation (317)). Either using an approximation for n
i
(T) such as an
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 Spring '09
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