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ECE 3080: Homework #2, Solution
June 4, 2009
Page 1 of 8
G
EORGIA
I
NSTITUTE OF
T
ECHNOLOGY
S
CHOOL OF
E
LECTRICAL AND
C
OMPUTER
E
NGINEERING
ECE 3080: Semiconductor Devices for Computer Engineering and
Telecommunication Systems
Summer Semester 2009, Homework #2
SOLUTION
1. Carrier Generation: Continuity Equation (50 points)
Assume a 1dimensional silicon crystal, which has been doped with 10
17
cm
3
phosphorous
atoms. Using a highenergy light source, 10
14
cm
3
additional electronhole pairs are
generated at x = 0 in the stationary state (dp/dt = dn/dt = 0).
(a) Can we consider lowlevel injection or do we have to assume highlevel injection?
Justify your answer.
(b) Calculate
analytical expressions
for the minority carrier distribution p
n
(x) in the
stationary state for x > 0 for the following two cases:
(i)
Infinitely long semiconductor with p
n
(x=
∞
) = p
n0
(see sketch (i))
(ii) Semiconductor with length W and p
n
(x=W) = p
n0
(see sketch (ii))
(c)
In case of the semiconductor with length W: give an approximate solution of the
minority carrier distribution in the case L
p
= (D
p
τ
p
)
1/2
>> W (hint: do a Taylor expansion
of the resulting sinh function, i.e. sinh(x)
≈
x, e
x
≈
1 + x, and e
x
≈
1 – x for small x)
Case (i):
Case (ii):
Comment: Use the 1D continuity equation, L
p
= (D
p
τ
p
)
1/2
, and sinh(x) = (e
x
 e
x
)/2.
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June 4, 2009
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Solution:
(a) Lowlevel injection can be considered, because the number of injected carriers
Δ
p =
Δ
n =
10
14
cm
3
is far smaller than the majority carrier density n
n0
= N
D
= 10
17
cm
3
. This way, the
majority carrier concentration remains basically unchanged, while the minority carrier
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