Georgia Tech
Lecture 5
Carrier Concentrations in Equilibrium
Reading:
(Cont’d) Notes and Anderson 2 sections from lecture 4
ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes
Motivation:
Since current (electron and hole flow) is dependent on the concentration of electrons and holes in the material, we need to develop equations that describe these concentrations.
Furthermore, we will find it useful to relate the these concentrations to the average energy (fermi energy) in the material.
Georgia Tech ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes
Probability the state is filled The density of electrons is: n
= ∫
E
E
Top
Bottom of of conduction conduction band band g c
( E ) f ( E ) dE
Number of states per cm -3 in energy range dE
The density of holes is:
Probability the state is empty p
= ∫
E
E
Top
Bottom of of valence band valence band g v
( E )[ 1
− f ( E )] dE
Number of states per cm -3 in energy range dE
Note: units of n and p are #/cm 3
Georgia Tech ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes n
= m n
*
2 m n
*
2 h 3
∫
E c
E
Top of conduction band
Letting when E
=
η =
E
−
E c
E c
,
η kT
=
0
Let and
E
Top of conduction band
→ ∞
1
+
E e
( E
−
−
E
E c f
) / kT
η c
=
E f
− kT
E c dE n
= m n
*
2 m n
*
( kT
2 h 3
)
3 / 2
0
∫ ∞
1
+
1 / 2 e
(
η − η c
) d
This is known as the Fermi-dirac integral of order 1/2 or, F
1/2
(
η c
)
ECE 3080 - Dr. Alan Doolittle Georgia Tech

Developing the mathematical model for electrons and holes
We can further define:
N c
=
⎡
2
⎣ m n
*
2
π
( kT h 2
)
⎤
⎥
3 / 2 and
N v
=
⎡
2
⎢
⎢
⎣ m
* p
2
π
( kT h 2
)
⎤
⎥
3 / 2 the effective density of states in the conduction band the effective density of states in the valence band
This is a general relationship holding for all materials and results in:
N c
=
2 .
51 x 10
19
⎛
⎜⎜ m m
* n o
⎞
⎟⎟
3
2 cm
−
3 at 300 K
Georgia Tech
N v
=
2 .
51 x 10
19
⎛
⎜⎜ m m o
* v
⎞
⎟⎟
3
2 cm
−
3 at 300 K
ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes n
=
N c
2
π
F
1 / 2
(
η c
) and p
=
N v
2
π
F
1 / 2
(
η v
) where
η v
=
( E v
−
E f
) / kT
Fermi-dirac integrals can be numerically determined or read from tables or...
Georgia Tech ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes
Useful approximations to the Fermi-dirac integral
If E f
< E c
-3kT
1
+
1 e
η − η c
≅ e
−
(
η − η c
)
F
1 / 2
(
c
)
=
2
e
( E f
−
E c
) / kT
De ge ne ra te
3kT
Ec
Ef
Similarly when E f
> E v
+3kT
F
1 / 2
Georgia Tech
(
v
)
=
2
e
( E v
−
E f
) / kT
Deg ene rate
3kT
Ev
ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes
Useful approximations to the Fermi-dirac integral:
Nondegenerate Case n
=
N c e
( E f
−
E c
) / kT and p
=
N v e
( E v
−
E f
) / kT
Georgia Tech ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes
When n=n i
, E f
=E i
(the intrinsic energy), then n i
=
N c e
( E i
−
E c
) / kT and n i
=
N v e
( E v
−
E i
) / kT or or
N c
N v
= n i e
( E c
−
E i
) / kT
= n i e
( E i
−
E v
) / kT n
= n i e
( E f
−
E i
) / kT and p
= n i e
( E i
−
E f
) / kT
Georgia Tech ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes
Other useful Relationships: n - p product n i
=
N c e
( E i
−
E c
) / kT n i
2 and
=
N c
N v e
−
( E c
−
E v
) / kT n i
= n i
=
N c
N v e
−
E
G
/ 2 kT
N v e
( E v
−
E i
=
N c
N v e
−
E
G
/ kT
) / kT
Georgia Tech ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes
Other useful Relationships: n - p product
Since n
= n i e
( E f
−
E i
) / kT and p
= n i e
( E i
−
E f
) / kT np
= n i
2
Known as the Law of mass Action
Georgia Tech ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes
Charge Neutrality
•If excess charge existed within the semiconductor, random motion of charge would imply net (AC) current flow. ===> Not possible!
•Thus, all charges within the semiconductor must cancel.
q
[ ( p
N
A
)
(
N d
n
) ]
0
Georgia Tech ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes
Charge Neutrality: Total Ionization Case
N -
A
= Concentration of “ionized” acceptors ~ = N
A
N +
D
= Concentration of “ionized” donors ~ = N
D
( p
N
A
) (
N
n
)
0
D
Georgia Tech ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes
Charge Neutrality: Total Ionization Case
( p
N N
n
A D
)
0
⎛
⎜⎜ n i
2 n n
2
−
N
A
⎞
⎟⎟
+
(
N
− n
)
=
0
D
n
(
N
D
N
A
)
n i
2
0
Watch out for round off error here!
(Use the form that has a positive first term combined with the law of mass action) n
=
N
D
−
N
A
2
+ ⎜
⎝
⎛
N
D
−
2
N
A
⎞
⎠
2
+ n i
2 or and pn
= n i
2 p
=
N
A
−
N
D
2
+ ⎜
⎝
⎛
N
A
−
2
N
D
⎞
⎠
2
+ n i
2
Georgia Tech ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes n
≅
If N
D
>>N
A and N
D
>>n i
N
D and p
≅ n i
2
N
D p
≅
If N
A
>>N
D and N
A
>>n i
N
A and n
≅ n i
2
N
A
Georgia Tech ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes
Example:
An intrinsic Silicon wafer has 1e10 cm -3 holes. When 1e18 cm -3 donors are added, what is the new hole concentration?
p
= n i
2 n n
=
≅
N
10
D
( )
2
=
10
18
10
18 cm
−
3 cm
−
3
=
100 cm
−
3
Georgia Tech ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes
Example:
An intrinsic Silicon wafer has 1e10 cm -3 holes. When 1e18 cm -3 acceptors and 8e17 cm -3 donors are added, what is the new hole concentration?
p
=
1 x 10
18 −
8 x 10
17
2
+
⎛
⎜⎜
1 x 10
18 −
8 x 10
17
2
⎞
⎟⎟
2 p
=
2 x 10
17 cm
−
3 =
N
A
−
N
D
+
(
1 x 10
10
2
)
Georgia Tech ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes
Example:
An intrinsic Silicon wafer at 470K has 1e14 cm -3 holes. When 1e14 cm -3 acceptors are added, what is the new electron and hole concentrations?
N
D
=0 p
=
1 x 10
14
2
+
⎛
⎜⎜
1 x 10
14
2
⎞
⎟⎟
2
+
(
1 x 10
14
2
) p n
=
=
1 .
62 x 10
14
(
1 x 10
14
2
)
1 .
62 x 10
14 cm
−
3
=
≠
N
A
−
N
D
6 .
2 x 10
13 cm
−
3
Georgia Tech ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes
Example:
An intrinsic Silicon wafer at 600K has 4e15 cm -3 holes. When 1e14 cm -3 acceptors are added, what is the new electron and hole concentrations?
N
D
=0 p
=
1 x 10
14
2
+
⎛
⎜⎜
1 x 10
14
2
⎞
⎟⎟
2
+
(
4 x 10
15
2
) p n
=
=
4
(
4 x 10
15 x 10
15 cm
2
)
=
4 x 10
15
−
3 = n i
≠
4 x 10
15
N
A cm
−
3
−
=
N
D n i
⇓
Intrinsic Material at High Temperatur e
Georgia Tech ECE 3080 - Dr. Alan Doolittle

Developing the mathematical model for electrons and holes
Temperature behavior of Doped Material
N-type
QuickTime Movie
P-type
QuickTime Movie
Georgia Tech ECE 3080 - Dr. Alan Doolittle