ECE 340
Lecture 9 : Carrier
Concentrations and the
Temperature Dependence
Class Outline:
• Intrinsic Carrier Concentrations
• Extrinsic Carrier Concentrations
•  Thermal Effects Revisited
Things you should know when you leave…
Key Questions
•  What is the charge neutrality
relationship?
•  How are the carrier concentrations
determined for extrinsic material?
•  What role does temperature play
in carrier concentrations in
extrinsic material?
•  What is compensated material?
M.J. Gilbert
ECE 340 – Lecture 9
09/17/12
Intrinsic Carrier Concentrations
We recall that by using the density of states and the Fermi function…
Etop
n=
For electrons:
∫ g (E ) f (E )dE
c
Ec
Ev
p=
For holes:
∫ g (E )[1 − f (E )]dE
We can simplify these
integrals to find closed
form relations…
v
Ebottom
3kbT
*
n b
⎡ m k T ⎤
N c = 2 ⎢
2 ⎥
2
π

⎣
⎦
*
p b
2
⎡ m k T ⎤
N v = 2 ⎢
⎥
⎢⎣ 2π ⎥⎦
3
2
3
2
3kbT
*
n, p
1 ⎞⎛ m
⎛
N c ,v = ⎜ 2.51 ×1019 3 ⎟⎜
cm ⎠⎜⎝ m0
⎝
M.J. Gilbert
⎞
⎟
⎟
⎠
3
2
(E f − E c )
n = Nce
k bT
(E v − E f )
ECE 340 – Lecture 9
p = Nve
k bT
09/17/12
Intrinsic Carrier Concentrations
Is this the best and easiest??
∞
F1/ 2 (ηc ) = ∫
0
η
1
2
η −ηc
1+ e
dη
2
2 ⎛ (E f − Ec ) ⎞
⎟⎟
F1/ 2 (ηc ) = η = ⎜⎜
3
3 ⎝ kbT ⎠
3
2
c
2
3
(E f − Ec )
F1/ 2 (ηc ) = e
k bT
Joyce-Dixon Approximation:
But we can use curve fitting (no
physics) to cover all of the ranges:
M.J. Gilbert
⎡ n
⎡ p
1 n ⎤
1 p ⎤
E f = Ec + kbT ⎢ln
+
=
E
−
k
T
ln
+
v
b ⎢
⎥
⎥
8 N c ⎦
8 N v ⎦
⎣ N c
⎣ N v
ECE 340 – Lecture 9
09/17/12
Intrinsic Carrier Concentrations
Recall that we can also find the dependence on temperature…
For intrinsic semiconductors, we know the following:
n = p = ni and Ei = Ef
Then the relations for n and p become:
ni = N c e
ni = N v e
Ei − Ec
k bT
Ev − Ei
k bT
N c = ni e
N v = ni e
Ec − Ei
k bT
Ei − Ev
k bT
By combining equations, we arrive at two very important and useful results:
E f − Ei
n = ni e
− Eg
k bT
Ei − E f
p = ni e kbT
M.J. Gilbert
ni = N c N v e 2 kbT
ECE 340 – Lecture 9
09/17/12
Extrinsic Carrier Concentrations
In most situations, we are dealing with extrinsic semiconductors…
Donor Doped Semiconductor
Band Diagram:
Δx
Ed
Ev
Acceptor Doped
Semiconductor Band Diagram:
Ea
Ev
M.J. Gilbert
ECE 340 – Lecture 9
09/17/12
Extrinsic Carrier Concentrations
Let’s consider a small separate parts of a uniformly doped semiconductor…
+
+
+
-
M.J. Gilbert
ECE 340 – Lecture 9
09/17/12
Extrinsic Carrier Concentrations
The preceding analysis leads to the charge neutrality relationship…
ch arg e
+
−
=
qp
−
qn
+
qN
−
qN
D
A
cm 3
= p − n + N D+ − N A− = 0
N = ND
+
D
N = NA
−
A
M.J. Gilbert
p − n − ND − N A = 0
ECE 340 – Lecture 9
09/17/12
Extrinsic Carrier Concentrations
Let’s use the charge neutrality relation…
p=
ni2
− n + ND − N A = 0
n
2
i
n
n
n 2 − n(N D − N A ) − ni2 = 0
2
⎤
N D − N A ⎡⎛ N D − N A ⎞
2
n=
+ ⎢⎜
⎟ + ni ⎥
2
2
⎠
⎢⎣⎝
⎥⎦
M.J. Gilbert
1
2
2
⎤
N A − N D ⎡⎛ N A − N D ⎞
2
p=
+ ⎢⎜
⎟ + ni ⎥
2
2
⎠
⎢⎣⎝
⎥⎦
ECE 340 – Lecture 9
1
2
09/17/12
Extrinsic Carrier Concentrations
The previous equations are generic, we can simplify in 4 cases…
NA = ND = 0
n = p = ni
n ≈ ND
ni2
p=
ND
or
p ≈ NA
ni2
n=
NA
n = p = ni
M.J. Gilbert
ECE 340 – Lecture 9
09/17/12
Extrinsic Carrier Concentrations
What about the 4th case? Compensated Semiconductors
+ + + + + + + -­‐ -­‐ -­‐ -­‐ -­‐ -­‐ -­‐ Sum of positive charges (holes and ionized donors) must balance sum
of negative charges (electrons and ionized acceptors): !
p0 +Nd+ = n0 +Na- M.J. Gilbert
ECE 340 – Lecture 9
09/17/12
Extrinsic Carrier Concentrations
Where is the intrinsic level, Ei, really located?
In intrinsic material, we know that n=p. Now use the Boltzmann equations:
Ei − E f
Nce
kbT
E f − Ei
= Nve
kbT
But,
*
N v ⎛⎜ m p ⎞⎟
=
N c ⎜⎝ mn* ⎟⎠
So,
3
2
*
⎛
⎞
m
Ec + Ev 3
p
Ei =
+ kbT ln⎜ * ⎟
⎜ m ⎟
2
4
⎝ n ⎠
M.J. Gilbert
ECE 340 – Lecture 9
09/17/12
Extrinsic Carrier Concentrations
So where does the Fermi level lie for a doped semiconductor??
E f − Ei
Solve for Ef – Ei:
n = ni e
Ei − E f
p = ni e
⎛ N D ⎞
⎟⎟
E f − Ei = kbT ln⎜⎜
⎝ ni ⎠
⎛ N A ⎞
⎟⎟
Ei − E f = kbT ln⎜⎜
⎝ ni ⎠
M.J. Gilbert
k bT
k bT
ND >> NA , ND >> ni
NA >> ND , NA >> ni
ECE 340 – Lecture 9
09/17/12
Extrinsic Carrier Concentrations
So where does the Fermi level lie for a doped semiconductor??
Fermi level positioning in Si at 300 K as a function of the doping concentration…
M.J. Gilbert
ECE 340 – Lecture 9
09/17/12
Thermal Effects Revisited
What is the role of temperature??
M.J. Gilbert
ECE 340 – Lecture 9
09/17/12