ECE 340
Homework X
Spring 2004
Due: Monday, March 15, 2004
1.
When a prolonged diffusion or a highenergy implantation is conducted to form a p/n junction. The doping profile near the
junction is usually graded, and the stepjunction approach is no longer suitable to find the relationship between the width of the
depletion region and the contact potential. However, the underlying principle used to establish equations 513 to 523 remains
intact, and they can still be used to determine similar equations for the graded junction.
Assume that the doping profile varies
as N
a
N
d
=Gx where G is 10
20
/cm
4
in a linear junction.
(a) Find and plot the electric field,
ε
(x),
for 
∞
<x<+
∞
.
(b) Determine the relationship between the width of the depletion region and contact potential for the junction at equilibrium.
Solutions:
(a), Please refer to section 5.6.4 and Figure 539 (page 220 to obtain a general understanding of the device.
We are going to use depletion approximation in the junction region and quasineutral approximation
outside junction region. The example in section 5.6.4 has N
d
N
a
=Gx, but we have N
a
N
d
=Gx.
Therefore, in our case, the
ρ
,
x
Ε
, and
V
will be the mirror image of those in figure 539 about the x
axis. After obtaining a general understanding of the device, we can solve the problem through steps
similar to equations 513 to 523:
Since it’s a symmetrical device,
0
0
/ 2
n
p
x
x
W
=
=
. For x outside depletion region (


/ 2
x
W
>
),
electrical field is 0.
For x within depletion region (


/ 2
x
W
<
), start with Poisson equation,
( )
( )
(
)
(
)
Depletion
d
a
d
a
approximation
d
x
q
d
x
q
q
p
n
N
N
N
N
Gx
dx
dx
ε
ε
ε
+
−
+
−
Ε
Ε
=
−
+
−
⎯⎯⎯⎯⎯→
=
−
= −
The general solution to above differential equation:
2
2
x
qG
x
Const
ε
Ε
= −
+
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