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9/11/12
1
ECE 340 Lecture 7 : Fermi Level and
Equilibrium Carrier Distributions
Class Outline:
• Density of States • Fermi Level • Equilibrium Carrier Distributions M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Things you should know when you leave…
Key Questions • What is the density of states? • What does the Fermi function tell
us? • What is the Fermi level? • How do I find the equilibrium
electron and hole concentrations? • When can I use the simplified
equations for the concentrations?
M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Extrinsic Materials
How tightly bound is the extra electron or hole?
r
Donor Acceptor
e-
h+
( ) 220
2
4*
32 r
nB
qmEεεπ
−=
Donor in Si P As Sb
Binding energy (eV) 0.045 0.054 0.039
Acceptor in Si B Al Ga In
Binding energy (eV) 0.045 0.067 0.072 0.16
M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Extrinsic Materials
Visualizing donors on the band diagram…
Ed
Ea Ev
Ec
Ev
Ec
Δx
Let’s take a look at Silicon with Phosphorus impurity atoms:
Ed
Ev
Ec
Eg = 1.12 eV
0.045 eV
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M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Density of States
We are counting states…
ky kx
kz
K
K + dK
Δk
( ) kKN D Δ= 33 82π
The total number of states in a small region of volume Δk can be written as (states / volume):
dkkk 24π=Δ
dEE
mdk
Emk
12
1
2
*
2
*
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛=
=
M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Density of States
But it is in k-space… The number of states in the shell is given in terms of the density of states in energy by :
( )dEEN
( ) ( )dkkdEEN d2
33 482
ππ
=
( ) EmmEN d*
32
*
3 2π
=
Ener
gy
Ec
Ev
Available Valence Band States
Available Conduction Band States
Eg
M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Density of States
Three things to know about the density of states (DOS)…
Ener
gy Ec
Ev
Eg
( ) ( )
( )( )32
**
32
**
2
2
π
π
EEmmEg
EEmmEg
vppv
cnnc
−=
−= E ≥ Ec
E ≤ Ev
Units are typically states/cm3-eV
M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Fermi Level
Fine, but how many states are ACTUALLY filled?
f(E)
1
0Ef E
( ) ( )TkEE
b
f
e
Ef −
+
=
1
1½
T = 0 K
Fermi Energy
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M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Fermi Level
What about the temperature dependence?
Ef + 3kbT
Ef - 3kbT
T = 300 K Ef = 0.5 eV
M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Fermi Level
Let’s look at a quick example…
f(E) 1 - f(E)
( ) ( )TkEEc
b
fc
e
Ef −
+
=
1
1 ( ) ( )
( )TkEE
TkEEv
b
vf
b
fv
e
e
Ef
−
−
+
+
−=−
1
11
111
TkEE
TkEE
b
vf
b
fc −=
−
2vc
fEEE +
=
Ec
Ev
Ef
M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Fermi Level
Visualizing the Fermi level in intrinsic material…
Ec
Ev
Ef
Ener
gy Ec
Ev
Eg Ef
Ener
gy Ec
Ev
Eg
Carrier Distributions:
M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Fermi Level
Visualizing the Fermi level in n-type material…
Ev
Ef
Ener
gy Ec
Ev
Eg
Ef
Ec
Ener
gy Ec
Ev
Eg
Carrier Distributions:
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M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Fermi Level
Visualizing the Fermi level in p-type material…
Ev
Ef
Ener
gy Ec
Ev
Eg Ef
Ec
Ener
gy Ec
Ev
Eg
Carrier Distributions:
M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Equilibrium Carrier Distributions
Let’s put together our knowledge of the density of states and the Fermi function…
We want to know how many carriers we have in our semiconductor for ANY energy or temperature.
∫ −
+
−=
top
c b
f
E
E TkEEcnn dE
e
EEmmn
1
232
**
π
( )TkEE
b
c−=η
( )TkEE
b
cfc
−=η Etop ~ ∞ ( )
ηη
π ηη de
Tkmmn
c
bnn ∫∞
−+=
0
21
32
23
**
1
M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Equilibrium Carrier Distributions
That integral is awful! Isn’t there an easier way?
( )η
ηπ ηη d
eTkmm
nc
bnn ∫∞
−+=
0
21
32
23
**
1
( )η
ηπ ηη d
eTkmm
pv
bpp∫∞
−+=
0
21
32
23
**
1
Fermi-Dirac integral of order ½.
Fermi-Dirac integral of order ½. Nc Nv
23
2
*
23
2
*
22
22
⎥⎥⎦
⎤
⎢⎢⎣
⎡=
⎥⎦
⎤⎢⎣
⎡=
π
π
TkmN
TkmN
bpv
bnc
23
0
*,
319
,11051.2 ⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎠
⎞⎜⎝
⎛ ×=mm
cmN pn
vc
( )
( )TkEE
v
TkEE
c
b
fv
b
cf
eNp
eNn−
−
=
=
M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Equilibrium Carrier Concentrations
These are simple closed form expressions, but are they the simplest?
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−
−
=
=
TkEE
i
TkEE
i
b
fi
b
if
enp
enn−
−
=
=TkE
vcib
g
eNNn 2−
=
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M.J. Gilbert ECE 340 – Lecture 7 09/12/12
Equilibrium Carrier Concentrations
A note on the effective mass…
kx
kz
ky
kx
ky
kz
ml
mt
mt
- - - - - - - - - - -
ml = 0.98 m0 mt = 0.19 m0
( ) 031
232
* 6 mmmm tln =
