ECE 340
Homework VIII
Spring 2004
Due: Friday, February 27, 2004
1.
A bar of silicon has a donor doping profile of n(x) = N
o
exp{Ax
2
} where A is a constant. Derive an expressing for the builtin
electric field as a function of x. If A = 5.6 x 10
7
/cm
2
, determine the position at which the electric field is 10
4
V/cm. Why is the
electric field zero at x=0?
Solutions:
Use quasineutral approximation to solve this problem. Quasineutral means that even there is carrier
redistribution due to diffusion or the induced drift, we assume the redistribution is small, therefore space
chargeneutrality holds everywhere. Under quasineutral assumption, we can approximate the carrier profile by
the doping profile.
At equilibrium, J = 0; Hole diffusion current can be neglected (why?), therefore J
p
= 0;
0
n
n
n
dn
J
q
n
qD
dx
µ
=
Ε +
=
, therefore:
n
n
D
dn
n dx
µ
Ε = −
By using Einstein relation (Equation. 429),
D
kT
q
µ
=
2
2
0
0
1
1
0.0259
0.0259
0.0259(2
)
0.0518
Ax
Ax
dN e
kT dn
dn
Ax
Ax
qn dx
n dx
dx
N e
−
−
Ε = −
= −
= −
=
=
If A = 5.6 x 10
7
/cm
2
, electric field is 10
4
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 Spring '08
 Staff
 Electric charge, 3 cm, 7.92 cm

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