ece 340 lecture 9 : carrier concentrations and the temperature...
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ECE 340 Lecture 9 : Carrier
Concentrations and the Temperature Dependence
Class Outline: • Intrinsic Carrier Concentrations • Extrinsic Carrier Concentrations • Thermal Effects Revisited
• 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
Things you should know when you leave…
Key Questions
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…
For electrons:
For holes:
( ) ( )∫=top
c
E
Ec dEEfEgn
( ) ( )[ ]∫ −=v
bottom
E
Ev dEEfEgp 1
We can simplify these integrals to find closed form relations…
23
2
*
23
2
*
22
22
⎥⎥⎦
⎤
⎢⎢⎣
⎡=
⎥⎦
⎤⎢⎣
⎡=
π
π
TkmN
TkmN
bpv
bnc
23
0
*,
319
,11051.2 ⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎠
⎞⎜⎝
⎛ ×=mm
cmN pn
vc
3kbT
3kbT ( )
( )TkEE
v
TkEE
c
b
fv
b
cf
eNp
eNn−
−
=
=
M.J. Gilbert ECE 340 – Lecture 9 09/17/12
Intrinsic Carrier Concentrations
Is this the best and easiest??
( ) ηη
η ηη de
Fcc ∫
∞
−+=0
21
2/1 1
( )( ) 3
2
23
2/1 32
32
⎟⎟⎠
⎞⎜⎜⎝
⎛ −==
TkEE
Fb
cfcc ηη
( )( )
TkEE
cb
cf
eF−
=η2/1
But we can use curve fitting (no physics) to cover all of the ranges: ⎥
⎦
⎤⎢⎣
⎡+−=⎥
⎦
⎤⎢⎣
⎡++=
vvbv
ccbcf N
pNpTkE
Nn
NnTkEE
81ln
81ln
Joyce-Dixon Approximation:
M.J. Gilbert 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:
TkEiE
vi
TkEE
ci
b
v
b
ci
eNn
eNn−
−
=
=
TkEE
iv
TkEE
ic
b
vi
b
ic
enN
enN−
−
=
=
By combining equations, we arrive at two very important and useful results:
TkEE
i
TkEE
i
b
fi
b
if
enp
enn−
−
=
=TkE
vcib
g
eNNn 2−
=
M.J. Gilbert ECE 340 – Lecture 9 09/17/12
Extrinsic Carrier Concentrations
In most situations, we are dealing with extrinsic semiconductors…
Ed
Ea
Ev
Ev
Δx
Donor Doped Semiconductor Band Diagram:
Acceptor Doped Semiconductor Band Diagram:
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…
0
arg3
=−+−=
−+−=
−+
−+
AD
AD
NNnp
qNqNqnqpcm
ech
0=−−− AD NNnpAA
DD
NNNN
=
=−
+
M.J. Gilbert ECE 340 – Lecture 9 09/17/12
Extrinsic Carrier Concentrations
Let’s use the charge neutrality relation…
nnp i2
= 02
=−+− ADi NNnnn
( ) 022 =−−− iAD nNNnn
21
22
22 ⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎠
⎞⎜⎝
⎛ −+
−= i
ADAD nNNNNn21
22
22 ⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎠
⎞⎜⎝
⎛ −+
−= i
DADA nNNNNp
M.J. Gilbert ECE 340 – Lecture 9 09/17/12
Extrinsic Carrier Concentrations
The previous equations are generic, we can simplify in 4 cases…
NA = ND = 0
n = p = ni
D
i
D
Nnp
Nn2
=
≈
A
i
A
Nnn
Np2
=
≈or
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:
TkEE
vTkEE
cb
if
b
fi
eNeN−−
=
But,
23
*
*
⎟⎟⎠
⎞⎜⎜⎝
⎛=
n
p
c
v
mm
NN
So,
⎟⎟⎠
⎞⎜⎜⎝
⎛+
+= *
*
ln43
2 n
pb
vci m
mTkEEE
M.J. Gilbert ECE 340 – Lecture 9 09/17/12
Extrinsic Carrier Concentrations
So where does the Fermi level lie for a doped semiconductor??
Solve for Ef – Ei: TkEE
i
TkEE
i
b
fi
b
if
enp
enn−
−
=
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=−
⎟⎟⎠
⎞⎜⎜⎝
⎛=−
i
Abfi
i
Dbif
nNTkEE
nNTkEE
ln
ln ND >> NA , ND >> ni
NA >> ND , NA >> ni
M.J. Gilbert 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??
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