Mark Lundstrom
10/5/13
ECE‐656
Fall 2013
1
SOLUTIONS:
ECE 656 Homework (Week 7)
Mark Lundstrom
Purdue University
(Revised 10/30/13)
1)
In Lecture 15, we derived a current equation for a 2D, n‐type conductor and wrote it as
J
n
=
σ
S
d F
n
q
(
)
dx
. Derive the corresponding equation for a p‐type semiconductor.
Solution:
I
=
2
q
h
T E
(
)
M
V
E
(
)
f
1
−
f
2
(
)
dE
−∞
E
V
∫
(channels in the valence band are all below
E
=
E
V
.)
f
1
−
f
2
≈
−
∂
f
1
∂
E
⎛
⎝
⎜
⎞
⎠
⎟
qV
T E
(
)
=
λ
λ
+
L
→
λ
L
I
=
2
q
h
λ
E
(
)
M
V
E
(
)
−
∂
f
0
∂
E
⎛
⎝
⎜
⎞
⎠
⎟
−∞
E
V
∫
dE
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
qV
L
J
px
=
−
I W
qV
=
−Δ
F
p
J
px
=
2
q
h
λ
E
(
)
M
V
E
(
)
W
(
)
−
∂
f
0
∂
E
⎛
⎝
⎜
⎞
⎠
⎟
−∞
E
V
∫
dE
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
Δ
F
p
L
J
px
=
2
q
h
λ
E
(
)
M
V
E
(
)
W
(
)
−
∂
f
0
∂
E
⎛
⎝
⎜
⎞
⎠
⎟
−∞
E
V
∫
dE
⎧
⎨
⎪
⎩
⎪
⎫
⎬
⎪
⎭
⎪
dF
p
dx
=
σ
Sp
dF
p
dx
J
px
=
σ
Sp
d F
p
q
(
)
dx
σ
Sp
=
2
q
2
h
λ
E
(
)
M
V
E
(
)
W
(
)
−
∂
f
0
∂
E
⎛
⎝
⎜
⎞
⎠
⎟
−∞
E
V
∫
dE
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Mark Lundstrom
10/5/13
ECE‐656
Fall 2013
2
2) In Lecture 15, we derived the drift‐diffusion equation for a 2D n‐type semiconductor
with parabolic energy bands. Repeat the derivation for a 3D semiconductor with
parabolic energy bands.
Do not
assume Maxwell‐Boltzmann statistics.
Solution:
Begin with:
J
nx
=
σ
dF
n
q
dx
(i)
n
=
N
C
F
1/2
η
F
(
)
η
F
=
F
n
−
E
C
(
)
k
B
T
N
C
=
1
4
2
m
*
k
B
T
π
2
⎛
⎝
⎜
⎞
⎠
⎟
3/2
Now find the gradient of the electrochemical potential:
dn
dx
=
N
C
d
d
η
F
F
1/2
η
F
(
)
⎧
⎨
⎩
⎪
⎫
⎬
⎭
⎪
d
η
F
dx
=
N
C
F
−
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